{"id":393,"date":"2016-06-01T20:49:47","date_gmt":"2016-06-01T20:49:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=393"},"modified":"2018-05-17T00:07:58","modified_gmt":"2018-05-17T00:07:58","slug":"outcome-solve-compound-inequalities-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/outcome-solve-compound-inequalities-2\/","title":{"raw":"Compound Inequalities","rendered":"Compound Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Describe sets as intersections or unions\r\n<ul>\r\n \t<li>Use interval notation to describe intersections and unions<\/li>\r\n \t<li>Use graphs to describe intersections and\u00a0unions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Solve compound inequalities\u2014OR\r\n<ul>\r\n \t<li>Solve compound inequalities in the form of <i>or<\/i> and express the solution graphically and with an interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Solve compound inequalities\u2014AND\r\n<ul>\r\n \t<li>Express solutions to inequalities graphically and with interval notation<\/li>\r\n \t<li>Identify solutions for compound inequalities in the form [latex]a&lt;x&lt;b[\/latex], including cases with no solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Use interval notation to describe sets of numbers as intersections and unions<\/h2>\r\nWhen two inequalities are joined by the word <i>and<\/i>, the solution of the simple compound inequality occurs when <i>both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word <i>or<\/i>, the solution of the compound inequality occurs when <i>either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.\r\n\r\nIn this section we will learn how to solve simple compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.\r\n\r\nVenn diagrams use the concept of intersections and unions to show how much two or more things share in common. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently Cecilia has both of these qualities; therefore she is the intersection of the two.\r\n\r\n<img class=\"aligncenter wp-image-3710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06222631\/Screen-Shot-2016-05-06-at-3.25.21-PM-300x234.png\" alt=\"Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.\" width=\"353\" height=\"275\" \/>\r\n\r\nIn mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers <em>x<\/em> can be, and what would the interval look like?\r\n\r\nIn words, <em>x<\/em> must be less than 6 <em>and at the same time,<\/em> it must be greater than 2, much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Let's look at a graph to see what numbers are possible with these constraints.\r\n\r\n<img class=\" wp-image-3958 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10234330\/Screen-Shot-2016-05-10-at-4.43.10-PM-300x46.png\" alt=\"x&gt; 2 and x&lt; 6\" width=\"594\" height=\"91\" \/>\r\n\r\nThe numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a <em>bounded<\/em> inequality and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. <em>x<\/em> must be less than 6 and greater than two\u2014the values for <em>x<\/em> will fall <em>between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:\r\n\r\n<img class=\"aligncenter wp-image-4014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/11235415\/Screen-Shot-2016-05-11-at-4.53.25-PM-300x46.png\" alt=\"Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.\" width=\"664\" height=\"102\" \/>\r\n\r\nOn the other hand, if you need to represent two things that\u00a0don't share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements but are often considered in the same application, such as online shopping or banking.\r\n\r\n<img class=\"aligncenter wp-image-3712\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06223115\/Screen-Shot-2016-05-06-at-3.26.52-PM-300x150.png\" alt=\"Two circles, one the Internet and the other Privacy.\" width=\"406\" height=\"203\" \/>\r\n\r\nIn mathematical terms, for example, [latex]x&gt;6[\/latex]\u00a0<em>or<\/em>\u00a0[latex]x&lt;2[\/latex] is an inequality joined by the word <em>or<\/em>. Using interval notation, we can describe each of these inequalities separately:\r\n\r\n[latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x&lt;2[\/latex] is the same as\u00a0[latex]\\left(\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0<em>or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0<em>or<\/em> real numbers greater than 6. Can you see why we need to write them as two separate intervals? Let's look at a graph to get a clear picture of what is going on.\r\n\r\n<img class=\"aligncenter wp-image-3960\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10235406\/Screen-Shot-2016-05-10-at-4.53.44-PM-300x39.png\" alt=\"Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.\" width=\"585\" height=\"76\" \/>\r\n\r\nWhen you place both of these inequalities on a graph, we can see that they share no numbers in common. This is what we call a union, as mentioned above. The interval notation associated with a union is a big U, so instead of writing <em>or<\/em>, we join our intervals with a big U, like this:\r\n\r\n[latex]\\left(\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]\r\n\r\nIt is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as\u00a0[latex]\\left(2,6\\right)[\/latex], where 2 is less than 6. The number on the right should be greater than the number on the left.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDraw the graph of the simple compound inequality [latex]x\\gt3[\/latex] <em>or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n\r\nNote, in interval notation, you are asked to find [latex](-\\infty,4] \\cup (3,\\infty)[\/latex].\r\n[reveal-answer q=\"641470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641470\"]\r\n\r\nThe graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on 3 and a blue arrow drawn to the right to contain all the numbers greater than 3.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182832\/image075.jpg\" alt=\"Number line. Open blue circle on 3. Blue highlight on all numbers greater than 3.\" width=\"575\" height=\"53\" \/>\r\n\r\nThe graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than 4.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182834\/image076.jpg\" alt=\"Number line. Closed red circle on 4. Red highlight on all numbers less than 4.\" width=\"575\" height=\"53\" \/>\r\n\r\nWhat do you notice about the graph that combines these two inequalities?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182836\/image077.jpg\" alt=\"Number line. Open blue circle on 3 and blue highlight on all numbers greater than 3. Red closed circle on 4 and red highlight through all numbers less than 4. This means that both colored highlights cover the numbers between 3 and 4.\" width=\"575\" height=\"53\" \/>\r\n\r\nSince this compound inequality is an <i>or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than 3 and less than or equal to 4 is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] or [latex]x\\le4[\/latex] is the set of all real numbers, and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex].\r\n\r\nAs a result we have: [latex](-\\infty,4] \\cup (3,\\infty)=(-\\infty,\\infty)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see two examples of how to express inequalities involving OR graphically and as an interval.\r\nhttps:\/\/youtu.be\/nKarzhZOFIk\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nDraw a graph of the simple compound inequality:\u00a0[latex]x\\lt5[\/latex]\u00a0<em><strong>and<\/strong><\/em>\u00a0[latex]x\\ge\u22121[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n\r\nNote, in interval notation, you are asked to find [latex](-\\infty,5) \\cap [-1,\\infty)[\/latex].\r\n[reveal-answer q=\"394627\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394627\"]\r\n\r\nThe graph of each individual inequality is shown in color.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182838\/image081.jpg\" alt=\"Number line. Open red circle on 5 and red arrow through all numbers less than 5. This red arrow is labeled x is less than 5. Closed blue circle on negative 1 and blue arrow through all numbers greater than negative 1. This blue arrow is labeled x is greater than or equal to negative 1.\" width=\"575\" height=\"53\" \/>\r\n\r\nSince the word <i>and <\/i>joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.\r\n\r\nThe solution to this compound inequality is shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182840\/image082.jpg\" alt=\"Number line. Closed blue circle on negative 1. Open red circle on 5. The numbers between negative 1 and 5 (including negative 1) are colored purple. The purple line is labeled negative 1 is less than or equal to x is less than 5.\" width=\"575\" height=\"53\" \/>\r\n\r\nNotice that this is a bounded inequality. You can rewrite [latex]x\\ge\u22121\\,\\text{and }x\\le5[\/latex] as [latex]\u22121\\le x\\le 5[\/latex]\u00a0since the solution is between [latex]\u22121[\/latex] and 5, including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex]\u00a0as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex]\u00a0<i>and<\/i> less than 5.\u201d You can rewrite an <i>and<\/i> statement this way only if the answer is <i>between<\/i> two numbers. The set of solutions to this inequality can be written in interval notation like this: [latex]\\left[{-1},{5}\\right)[\/latex]\r\n\r\nAs a result we have: [latex](-\\infty,5)) \\cap [-1,\\infty)=[-1,5)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nDraw the graph of the simple compound inequality [latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n\r\nNote, in interval notation, you are asked to find [latex](-\\infty,-3) \\cap (3,\\infty)[\/latex].\r\n[reveal-answer q=\"870500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"870500\"]\r\n\r\nFirst, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both\u00a0<\/strong>inequalities since they are joined with the word <em>and<\/em>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/>\r\n\r\nIn this case, there are no shared <em>x<\/em>-values, and therefore there is no intersection for these two inequalities. We can write \"no solution,\" or DNE.\r\n\r\nAs a result we have:\u00a0[latex](-\\infty,-3) \\cap (3,\\infty) = \\varnothing[\/latex] (DNE).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video presents two examples of how to draw inequalities involving AND, as well as write the corresponding intervals.\r\n\r\nhttps:\/\/youtu.be\/LP3fsZNjJkc\r\n<h2>Solve general compound inequalities in the form of <i>or<\/i><\/h2>\r\nAs we saw in the last section, the solution of a simple compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.\r\n\r\nIn this section you will see that some inequalities need to be simplified before their solution can be written or graphed.\r\n\r\nIn the following example, you will see an example of how to solve a one-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>. \u00a0[latex]3x\u20131&lt;8[\/latex] or [latex]x\u20135&gt;0[\/latex]\r\n\r\n[reveal-answer q=\"212910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"212910\"]Solve each inequality by isolating the variable.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}x-5&gt;0\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1&lt;8\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,+1}\\\\x\\,\\,\\,\\,\\,\\,&gt;5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,&lt;\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x&lt;3\\,\\,\\,\\\\x&gt;5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x&lt;3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nWrite both inequality solutions as a compound using <i>or,\u00a0<\/i>using interval notation.\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle x&gt;5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x&lt;3[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]\r\n\r\nThe solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first before writing the solution using interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182844\/image078.jpg\" alt=\"Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <em>y<\/em>.\u00a0\u00a0[latex]2y+7\\lt13\\text{ or }\u22123y\u20132\\lt10[\/latex]\r\n[reveal-answer q=\"969462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"969462\"]\r\n\r\nSolve each inequality separately.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}2y+7&lt;13\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-3y-2\\le 10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,\\,\\,+2}\\\\\\frac{2y}{2}\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,\\,\\frac{6}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-3y}{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{12}{-3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y&lt;3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\ge -4\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y&lt;3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,y\\ge -4\\end{array}[\/latex]<\/p>\r\nThe inequality sign is reversed with division by a negative number.\r\n\r\nSince <i>y<\/i> could be less than 3 or greater than or equal to [latex]\u22124[\/latex], <i>y<\/i> could be any number. Graphing the inequality helps with this interpretation.\r\n<h4>Answer<\/h4>\r\nInequality: [latex]y&lt;3\\text{ or }y\\ge -4[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182734\/image079.jpg\" alt=\"Closed dot on negative 4 and shaded line going through all numbers greater than negative 4. Open dot on 3 and shaded line on all numbers less than 3. Numbers between closed dot on negative 4 and open dot on 3 are shaded twice.\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution \"all real numbers.\" \u00a0Any real number will produce a true statement for either\u00a0[latex]y&lt;3\\text{ or }y\\ge -4[\/latex], when it is substituted for <em>x<\/em>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <em>z<\/em>.\u00a0[latex]5z\u20133\\gt\u221218[\/latex] or [latex]\u22122z\u20131\\gt15[\/latex]\r\n[reveal-answer q=\"74043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"74043\"]\r\n\r\nSolve each inequality separately.\u00a0Combine the solutions.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}5z-3&gt;-18\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-2z-1&gt;15\\\\\\underline{\\,\\,\\,\\,\\,\\,+3\\,\\,\\,\\,\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,+1}\\\\\\frac{5z}{5}\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\,\\frac{-15}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-2z}{-2}\\,\\,\\,\\,\\,\\,&lt;\\,\\,\\frac{16}{-2}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z&gt;-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z&lt;-8\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z&gt;-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z&lt;-8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality:\u00a0[latex] \\displaystyle z&gt;-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z&lt;-8[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z&lt;-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x&gt;-3[\/latex] has solutions that continue all the way to the right. In this way we write solutions with intervals from left to right.\r\n\r\nGraph:<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182846\/image080.jpg\" alt=\"Number line. Red open circle on negative 8 and red highlight on all numbers less than negative 8. Open blue circle on negative 3 and blue highlight through all numbers greater than negative 3.\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains an example of solving a compound inequality involving OR, and drawing the associated graph.\r\n\r\nhttps:\/\/youtu.be\/oRlJ8G7trR8\r\n\r\nIn the next section you will see examples of how to solve compound inequalities containing <em>and<\/em>.\r\n<h2>Solve general compound inequalities in the form of <i>and<\/i> and express the solution graphically<\/h2>\r\nThe solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this is\u00a0where the two graphs overlap.\r\n\r\nIn this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\u00a0[latex] \\displaystyle 1-4x\\le 21\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,5x+2\\ge22[\/latex]\r\n\r\n[reveal-answer q=\"266032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266032\"]\r\n\r\nSolve each inequality for <em>x<\/em>.\u00a0Determine the intersection of the solutions.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,1-4x\\le 21\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,5x+2\\ge 22\\\\\\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,-2}\\\\\\,\\,\\,\\,\\,\\underline{-4x}\\leq \\underline{20}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{5x}\\,\\,\\,\\,\\,\\,\\,\\ge \\underline{20}\\\\\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge -5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 4\\,\\,\\,\\,\\\\\\\\x\\ge -5\\,\\text{and}\\,\\,x\\ge 4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThe number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\\geq4[\/latex], since\u00a0this is where the two graphs overlap.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182847\/image083.jpg\" alt=\"Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/>\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle x\\ge 4[\/latex]\r\n\r\nInterval: [latex]\\left[4,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/image084.jpg\" alt=\"Number line. Closed purple circle (overlapping red and blue circles) on 4 and purple arrow through all numbers greater than 4. Purple line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>: \u00a0[latex] \\displaystyle {5}{x}-{2}\\le{3}\\text{ and }{4}{x}{+7}&gt;{3}[\/latex]\r\n[reveal-answer q=\"784358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"784358\"]\r\n\r\nSolve each inequality separately.\u00a0Find the overlap between the solutions.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,4x+7&gt;\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\frac{5x}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{5}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{4x}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\frac{-4}{4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;-1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x&gt;-1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]-1\\le{x}\\le{1}[\/latex]\r\n\r\nInterval: [latex]\\left(-1,1\\right)[\/latex]\r\n\r\nGraph:<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182704\/image085.jpg\" alt=\"image\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Compound inequalities in the form [latex]a&lt;x&lt;b[\/latex]<\/h2>\r\nRather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a&lt;x&lt;b[\/latex]\u00a0into two inequalities [latex]x&lt;b[\/latex]<i> and <\/i>[latex]x&gt;a[\/latex], you can more quickly to solve the inequality by applying the properties of inequality to all three segments of the compound inequality.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>. [latex]3\\lt2x+3\\leq 7[\/latex]\r\n\r\n[reveal-answer q=\"39150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"39150\"]\r\n\r\nIsolate the variable by subtracting 3 from all 3 parts of the inequality, then dividing each part by 2.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,3\\,\\,\\lt\\,\\,2x+3\\,\\,\\leq \\,\\,\\,\\,7\\\\\\underline{\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-3}\\,\\\\\\,\\,\\,\\,\\,\\underline{\\,0\\,}\\,\\,\\lt\\,\\,\\,\\,\\underline{2x}\\,\\,\\,\\,\\,\\,\\,\\,\\leq\\,\\,\\,\\underline{\\,4\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\lt x\\leq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle 0\\lt{x}\\le 2[\/latex]\r\n\r\nInterval: [latex]\\left(0,2\\right][\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter wp-image-3962\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182851\/Screen-Shot-2016-05-10-at-4.58.30-PM-300x77.png\" alt=\"Open dot on zero, closed dot on 2, and line through all numbers between zero and two.\" width=\"366\" height=\"94\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\nIn the video below, you will see another example of how to solve an inequality in the form \u00a0[latex]a&lt;x&lt;b[\/latex]\r\n\r\nhttps:\/\/youtu.be\/UU_KJI59_08\r\n<div class=\"textbox shaded\">\r\n<h3>Example 8: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\r\nSolve the compound inequality with variables in all three parts: [latex]3+x&gt;7x - 2&gt;5x - 10[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Solution<\/h3>\r\nLets try the first method. Write two inequalities<strong>:<\/strong>\r\n<div style=\"text-align: center\">[latex]\\begin{array}{lll}3+x&gt; 7x - 2\\hfill &amp; \\text{and}\\hfill &amp; 7x - 2&gt; 5x - 10\\hfill \\\\ 3&gt; 6x - 2\\hfill &amp; \\hfill &amp; 2x - 2&gt; -10\\hfill \\\\ 5&gt; 6x\\hfill &amp; \\hfill &amp; 2x&gt; -8\\hfill \\\\ \\frac{5}{6}&gt; x\\hfill &amp; \\hfill &amp; x&gt; -4\\hfill \\\\ x&lt; \\frac{5}{6}\\hfill &amp; \\hfill &amp; -4&lt; x\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is [latex]-4&lt;x&lt;\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200410\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n<\/div>\r\nTo solve inequalities like [latex]a&lt;x&lt;b[\/latex], use the addition and multiplication properties of inequality to solve the inequality for <i>x<\/i>. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.\r\n\r\nThe solution to a compound inequality with <i>and<\/i> is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word <i>and<\/i>:\r\n<table width=\"535\">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\">Case 1:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Description<\/td>\r\n<td>The solution could be all the values between two endpoints<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Inequalities<\/td>\r\n<td>[latex]x\\le{1}[\/latex] and [latex]x\\gt{-1}[\/latex], or as a bounded inequality: [latex]{-1}\\lt{x}\\le{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Interval<\/td>\r\n<td>[latex](-\\infty,1] \\cap (-1,\\infty) = \\left(-1,1\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Graphs<\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064043\/image089.jpg\" alt=\"Number line. Open blue circle on negative 1 and blue arrow through all numbers greater than negative 1. The blue arrow represents x is greater than negative 1. Closed red circle on 1 and red arrow through all numbers less than 1. Red arrow written x is less than or equal to 1.\" width=\"575\" height=\"53\" \/>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064015\/image085.jpg\" alt=\"Number line. Open blue circle on negative 1. Closed red circle on 1. Overlapping red and blue lines between negative 1 and 1 that represents negative 1 is less than x is less than or equal to 1.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<th colspan=\"2\">Case 2:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Description<\/td>\r\n<td>The solution could begin at a point on the number line and extend in one direction.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Inequalities<\/td>\r\n<td>[latex]x\\gt3[\/latex] and [latex]x\\ge4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Interval<\/td>\r\n<td>[latex](-3,\\infty) \\cap [4,\\infty) = \\left[4,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Graphs<\/td>\r\n<td>\u00a0<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064045\/image090.jpg\" alt=\"Number line. Blue open circle on negative 3 and blue arrow through all numbers greater than negative 3. Blue arrow represents x is greater than negative three. Closed red circle on 4 and red arrow through all numbers greater than 4. The red arrow respresents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"Number line. Closed circle on 4 and arrow through all numbers greater than 4. The arrow represents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<th colspan=\"2\">Case 3:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Description<\/td>\r\n<td>In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Inequalities<\/td>\r\n<td>[latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Intervals<\/td>\r\n<td>[latex]\\left(-\\infty,-3\\right) \\cap \\left(3,\\infty\\right) = \\varnothing[\/latex] (DNE)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Graph<\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064046\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>.\u00a0[latex]x+2&gt;5[\/latex] and [latex]x+4&lt;5[\/latex]\r\n\r\n[reveal-answer q=\"336256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"336256\"]\r\n\r\nSolve each inequality separately.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}x+2&gt;5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,x+4&lt;5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,&lt;\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;3\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x&lt;1\\end{array}[\/latex]<\/p>\r\nFind the overlap between the solutions.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182858\/image092.jpg\" alt=\"Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/>\r\n<h4>Answer<\/h4>\r\nThere is no overlap between [latex] \\displaystyle x&gt;3[\/latex] and [latex]x&lt;1[\/latex], so there is no solution.[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Describe sets as intersections or unions\n<ul>\n<li>Use interval notation to describe intersections and unions<\/li>\n<li>Use graphs to describe intersections and\u00a0unions<\/li>\n<\/ul>\n<\/li>\n<li>Solve compound inequalities\u2014OR\n<ul>\n<li>Solve compound inequalities in the form of <i>or<\/i> and express the solution graphically and with an interval<\/li>\n<\/ul>\n<\/li>\n<li>Solve compound inequalities\u2014AND\n<ul>\n<li>Express solutions to inequalities graphically and with interval notation<\/li>\n<li>Identify solutions for compound inequalities in the form [latex]a<x<b[\/latex], including cases with no solution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Use interval notation to describe sets of numbers as intersections and unions<\/h2>\n<p>When two inequalities are joined by the word <i>and<\/i>, the solution of the simple compound inequality occurs when <i>both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word <i>or<\/i>, the solution of the compound inequality occurs when <i>either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.<\/p>\n<p>In this section we will learn how to solve simple compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.<\/p>\n<p>Venn diagrams use the concept of intersections and unions to show how much two or more things share in common. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently Cecilia has both of these qualities; therefore she is the intersection of the two.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06222631\/Screen-Shot-2016-05-06-at-3.25.21-PM-300x234.png\" alt=\"Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.\" width=\"353\" height=\"275\" \/><\/p>\n<p>In mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers <em>x<\/em> can be, and what would the interval look like?<\/p>\n<p>In words, <em>x<\/em> must be less than 6 <em>and at the same time,<\/em> it must be greater than 2, much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Let&#8217;s look at a graph to see what numbers are possible with these constraints.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3958 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10234330\/Screen-Shot-2016-05-10-at-4.43.10-PM-300x46.png\" alt=\"x&gt; 2 and x&lt; 6\" width=\"594\" height=\"91\" \/><\/p>\n<p>The numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a <em>bounded<\/em> inequality and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. <em>x<\/em> must be less than 6 and greater than two\u2014the values for <em>x<\/em> will fall <em>between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/11235415\/Screen-Shot-2016-05-11-at-4.53.25-PM-300x46.png\" alt=\"Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.\" width=\"664\" height=\"102\" \/><\/p>\n<p>On the other hand, if you need to represent two things that\u00a0don&#8217;t share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements but are often considered in the same application, such as online shopping or banking.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3712\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06223115\/Screen-Shot-2016-05-06-at-3.26.52-PM-300x150.png\" alt=\"Two circles, one the Internet and the other Privacy.\" width=\"406\" height=\"203\" \/><\/p>\n<p>In mathematical terms, for example, [latex]x>6[\/latex]\u00a0<em>or<\/em>\u00a0[latex]x<2[\/latex] is an inequality joined by the word <em>or<\/em>. Using interval notation, we can describe each of these inequalities separately:<\/p>\n<p>[latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x<2[\/latex] is the same as\u00a0[latex]\\left(\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0<em>or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0<em>or<\/em> real numbers greater than 6. Can you see why we need to write them as two separate intervals? Let&#8217;s look at a graph to get a clear picture of what is going on.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3960\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10235406\/Screen-Shot-2016-05-10-at-4.53.44-PM-300x39.png\" alt=\"Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.\" width=\"585\" height=\"76\" \/><\/p>\n<p>When you place both of these inequalities on a graph, we can see that they share no numbers in common. This is what we call a union, as mentioned above. The interval notation associated with a union is a big U, so instead of writing <em>or<\/em>, we join our intervals with a big U, like this:<\/p>\n<p>[latex]\\left(\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]<\/p>\n<p>It is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as\u00a0[latex]\\left(2,6\\right)[\/latex], where 2 is less than 6. The number on the right should be greater than the number on the left.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Draw the graph of the simple compound inequality [latex]x\\gt3[\/latex] <em>or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<p>Note, in interval notation, you are asked to find [latex](-\\infty,4] \\cup (3,\\infty)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641470\">Show Solution<\/span><\/p>\n<div id=\"q641470\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on 3 and a blue arrow drawn to the right to contain all the numbers greater than 3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182832\/image075.jpg\" alt=\"Number line. Open blue circle on 3. Blue highlight on all numbers greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<p>The graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than 4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182834\/image076.jpg\" alt=\"Number line. Closed red circle on 4. Red highlight on all numbers less than 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p>What do you notice about the graph that combines these two inequalities?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182836\/image077.jpg\" alt=\"Number line. Open blue circle on 3 and blue highlight on all numbers greater than 3. Red closed circle on 4 and red highlight through all numbers less than 4. This means that both colored highlights cover the numbers between 3 and 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Since this compound inequality is an <i>or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than 3 and less than or equal to 4 is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] or [latex]x\\le4[\/latex] is the set of all real numbers, and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex].<\/p>\n<p>As a result we have: [latex](-\\infty,4] \\cup (3,\\infty)=(-\\infty,\\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see two examples of how to express inequalities involving OR graphically and as an interval.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solutions to Basic OR Compound Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nKarzhZOFIk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Draw a graph of the simple compound inequality:\u00a0[latex]x\\lt5[\/latex]\u00a0<em><strong>and<\/strong><\/em>\u00a0[latex]x\\ge\u22121[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<p>Note, in interval notation, you are asked to find [latex](-\\infty,5) \\cap [-1,\\infty)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394627\">Show Solution<\/span><\/p>\n<div id=\"q394627\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of each individual inequality is shown in color.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182838\/image081.jpg\" alt=\"Number line. Open red circle on 5 and red arrow through all numbers less than 5. This red arrow is labeled x is less than 5. Closed blue circle on negative 1 and blue arrow through all numbers greater than negative 1. This blue arrow is labeled x is greater than or equal to negative 1.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Since the word <i>and <\/i>joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.<\/p>\n<p>The solution to this compound inequality is shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182840\/image082.jpg\" alt=\"Number line. Closed blue circle on negative 1. Open red circle on 5. The numbers between negative 1 and 5 (including negative 1) are colored purple. The purple line is labeled negative 1 is less than or equal to x is less than 5.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Notice that this is a bounded inequality. You can rewrite [latex]x\\ge\u22121\\,\\text{and }x\\le5[\/latex] as [latex]\u22121\\le x\\le 5[\/latex]\u00a0since the solution is between [latex]\u22121[\/latex] and 5, including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex]\u00a0as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex]\u00a0<i>and<\/i> less than 5.\u201d You can rewrite an <i>and<\/i> statement this way only if the answer is <i>between<\/i> two numbers. The set of solutions to this inequality can be written in interval notation like this: [latex]\\left[{-1},{5}\\right)[\/latex]<\/p>\n<p>As a result we have: [latex](-\\infty,5)) \\cap [-1,\\infty)=[-1,5)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Draw the graph of the simple compound inequality [latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<p>Note, in interval notation, you are asked to find [latex](-\\infty,-3) \\cap (3,\\infty)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q870500\">Show Solution<\/span><\/p>\n<div id=\"q870500\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both\u00a0<\/strong>inequalities since they are joined with the word <em>and<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<p>In this case, there are no shared <em>x<\/em>-values, and therefore there is no intersection for these two inequalities. We can write &#8220;no solution,&#8221; or DNE.<\/p>\n<p>As a result we have:\u00a0[latex](-\\infty,-3) \\cap (3,\\infty) = \\varnothing[\/latex] (DNE).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video presents two examples of how to draw inequalities involving AND, as well as write the corresponding intervals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solutions to Basic AND Compound Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LP3fsZNjJkc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Solve general compound inequalities in the form of <i>or<\/i><\/h2>\n<p>As we saw in the last section, the solution of a simple compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.<\/p>\n<p>In this section you will see that some inequalities need to be simplified before their solution can be written or graphed.<\/p>\n<p>In the following example, you will see an example of how to solve a one-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>. \u00a0[latex]3x\u20131<8[\/latex] or [latex]x\u20135>0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q212910\">Show Solution<\/span><\/p>\n<div id=\"q212910\" class=\"hidden-answer\" style=\"display: none\">Solve each inequality by isolating the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}x-5>0\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1<8\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,+1}\\\\x\\,\\,\\,\\,\\,\\,>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,<\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x<3\\,\\,\\,\\\\x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Write both inequality solutions as a compound using <i>or,\u00a0<\/i>using interval notation.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3[\/latex]\n\nInterval: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]\n\nThe solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first before writing the solution using interval notation.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182844\/image078.jpg\" alt=\"Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <em>y<\/em>.\u00a0\u00a0[latex]2y+7\\lt13\\text{ or }\u22123y\u20132\\lt10[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q969462\">Show Solution<\/span><\/p>\n<div id=\"q969462\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}2y+7<13\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-3y-2\\le 10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,\\,\\,+2}\\\\\\frac{2y}{2}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\frac{6}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-3y}{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{12}{-3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\ge -4\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,y\\ge -4\\end{array}[\/latex]<\/p>\n<p>The inequality sign is reversed with division by a negative number.<\/p>\n<p>Since <i>y<\/i> could be less than 3 or greater than or equal to [latex]\u22124[\/latex], <i>y<\/i> could be any number. Graphing the inequality helps with this interpretation.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]y<3\\text{ or }y\\ge -4[\/latex]\n\nInterval: [latex]\\left(-\\infty,\\infty\\right)[\/latex]\n\nGraph:\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182734\/image079.jpg\" alt=\"Closed dot on negative 4 and shaded line going through all numbers greater than negative 4. Open dot on 3 and shaded line on all numbers less than 3. Numbers between closed dot on negative 4 and open dot on 3 are shaded twice.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution &#8220;all real numbers.&#8221; \u00a0Any real number will produce a true statement for either\u00a0[latex]y<3\\text{ or }y\\ge -4[\/latex], when it is substituted for <em>x<\/em>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <em>z<\/em>.\u00a0[latex]5z\u20133\\gt\u221218[\/latex] or [latex]\u22122z\u20131\\gt15[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q74043\">Show Solution<\/span><\/p>\n<div id=\"q74043\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.\u00a0Combine the solutions.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}5z-3>-18\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-2z-1>15\\\\\\underline{\\,\\,\\,\\,\\,\\,+3\\,\\,\\,\\,\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,+1}\\\\\\frac{5z}{5}\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\frac{-15}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-2z}{-2}\\,\\,\\,\\,\\,\\,<\\,\\,\\frac{16}{-2}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z<-8\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality:\u00a0[latex]\\displaystyle z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8[\/latex]\n\nInterval: [latex]\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[\/latex] has solutions that continue all the way to the right. In this way we write solutions with intervals from left to right.<\/p>\n<p>Graph:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182846\/image080.jpg\" alt=\"Number line. Red open circle on negative 8 and red highlight on all numbers less than negative 8. Open blue circle on negative 3 and blue highlight through all numbers greater than negative 3.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains an example of solving a compound inequality involving OR, and drawing the associated graph.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Solve a Compound Inequality Involving OR (Union)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/oRlJ8G7trR8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next section you will see examples of how to solve compound inequalities containing <em>and<\/em>.<\/p>\n<h2>Solve general compound inequalities in the form of <i>and<\/i> and express the solution graphically<\/h2>\n<p>The solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this is\u00a0where the two graphs overlap.<\/p>\n<p>In this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.\u00a0[latex]\\displaystyle 1-4x\\le 21\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,5x+2\\ge22[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266032\">Show Solution<\/span><\/p>\n<div id=\"q266032\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality for <em>x<\/em>.\u00a0Determine the intersection of the solutions.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,1-4x\\le 21\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,5x+2\\ge 22\\\\\\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,-2}\\\\\\,\\,\\,\\,\\,\\underline{-4x}\\leq \\underline{20}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{5x}\\,\\,\\,\\,\\,\\,\\,\\ge \\underline{20}\\\\\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge -5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 4\\,\\,\\,\\,\\\\\\\\x\\ge -5\\,\\text{and}\\,\\,x\\ge 4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>The number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\\geq4[\/latex], since\u00a0this is where the two graphs overlap.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182847\/image083.jpg\" alt=\"Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle x\\ge 4[\/latex]<\/p>\n<p>Interval: [latex]\\left[4,\\infty\\right)[\/latex]<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/image084.jpg\" alt=\"Number line. Closed purple circle (overlapping red and blue circles) on 4 and purple arrow through all numbers greater than 4. Purple line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>: \u00a0[latex]\\displaystyle {5}{x}-{2}\\le{3}\\text{ and }{4}{x}{+7}>{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q784358\">Show Solution<\/span><\/p>\n<div id=\"q784358\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.\u00a0Find the overlap between the solutions.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,4x+7>\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\frac{5x}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{5}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{4x}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\frac{-4}{4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>-1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x>-1\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]-1\\le{x}\\le{1}[\/latex]<\/p>\n<p>Interval: [latex]\\left(-1,1\\right)[\/latex]<\/p>\n<p>Graph:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182704\/image085.jpg\" alt=\"image\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Compound inequalities in the form [latex]a<x<b[\/latex]<\/h2>\n<p>Rather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a<x<b[\/latex]\u00a0into two inequalities [latex]x<b[\/latex]<i> and <\/i>[latex]x>a[\/latex], you can more quickly to solve the inequality by applying the properties of inequality to all three segments of the compound inequality.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>. [latex]3\\lt2x+3\\leq 7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q39150\">Show Solution<\/span><\/p>\n<div id=\"q39150\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the variable by subtracting 3 from all 3 parts of the inequality, then dividing each part by 2.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\,\\,\\,\\,3\\,\\,\\lt\\,\\,2x+3\\,\\,\\leq \\,\\,\\,\\,7\\\\\\underline{\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-3}\\,\\\\\\,\\,\\,\\,\\,\\underline{\\,0\\,}\\,\\,\\lt\\,\\,\\,\\,\\underline{2x}\\,\\,\\,\\,\\,\\,\\,\\,\\leq\\,\\,\\,\\underline{\\,4\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\lt x\\leq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle 0\\lt{x}\\le 2[\/latex]<\/p>\n<p>Interval: [latex]\\left(0,2\\right][\/latex]<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3962\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182851\/Screen-Shot-2016-05-10-at-4.58.30-PM-300x77.png\" alt=\"Open dot on zero, closed dot on 2, and line through all numbers between zero and two.\" width=\"366\" height=\"94\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<p>In the video below, you will see another example of how to solve an inequality in the form \u00a0[latex]a<x<b[\/latex]\n\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Solve a Compound Inequality Involving AND (Intersection)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UU_KJI59_08?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>Example 8: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\n<p>Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\n<p>Lets try the first method. Write two inequalities<strong>:<\/strong><\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\text{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\frac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\frac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is [latex]-4<x<\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.\n\n\n\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200410\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<\/div>\n<p>To solve inequalities like [latex]a<x<b[\/latex], use the addition and multiplication properties of inequality to solve the inequality for <i>x<\/i>. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.<\/p>\n<p>The solution to a compound inequality with <i>and<\/i> is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word <i>and<\/i>:<\/p>\n<table style=\"width: 535px;\">\n<tbody>\n<tr>\n<th colspan=\"2\">Case 1:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">Description<\/td>\n<td>The solution could be all the values between two endpoints<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Inequalities<\/td>\n<td>[latex]x\\le{1}[\/latex] and [latex]x\\gt{-1}[\/latex], or as a bounded inequality: [latex]{-1}\\lt{x}\\le{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Interval<\/td>\n<td>[latex](-\\infty,1] \\cap (-1,\\infty) = \\left(-1,1\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Graphs<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064043\/image089.jpg\" alt=\"Number line. Open blue circle on negative 1 and blue arrow through all numbers greater than negative 1. The blue arrow represents x is greater than negative 1. Closed red circle on 1 and red arrow through all numbers less than 1. Red arrow written x is less than or equal to 1.\" width=\"575\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064015\/image085.jpg\" alt=\"Number line. Open blue circle on negative 1. Closed red circle on 1. Overlapping red and blue lines between negative 1 and 1 that represents negative 1 is less than x is less than or equal to 1.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2\">Case 2:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">Description<\/td>\n<td>The solution could begin at a point on the number line and extend in one direction.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Inequalities<\/td>\n<td>[latex]x\\gt3[\/latex] and [latex]x\\ge4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Interval<\/td>\n<td>[latex](-3,\\infty) \\cap [4,\\infty) = \\left[4,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Graphs<\/td>\n<td>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064045\/image090.jpg\" alt=\"Number line. Blue open circle on negative 3 and blue arrow through all numbers greater than negative 3. Blue arrow represents x is greater than negative three. Closed red circle on 4 and red arrow through all numbers greater than 4. The red arrow respresents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"Number line. Closed circle on 4 and arrow through all numbers greater than 4. The arrow represents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2\">Case 3:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Description<\/td>\n<td>In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Inequalities<\/td>\n<td>[latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Intervals<\/td>\n<td>[latex]\\left(-\\infty,-3\\right) \\cap \\left(3,\\infty\\right) = \\varnothing[\/latex] (DNE)<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Graph<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064046\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>.\u00a0[latex]x+2>5[\/latex] and [latex]x+4<5[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q336256\">Show Solution<\/span><\/p>\n<div id=\"q336256\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}x+2>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,x+4<5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,<\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>3\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x<1\\end{array}[\/latex]<\/p>\n<p>Find the overlap between the solutions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182858\/image092.jpg\" alt=\"Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<h4>Answer<\/h4>\n<p>There is no overlap between [latex]\\displaystyle x>3[\/latex] and [latex]x<1[\/latex], so there is no solution.<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-393\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Screenshot: Cecilia Venn Diagram. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Internet Privacy. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solutions to Basic OR Compound Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nKarzhZOFIk\">https:\/\/youtu.be\/nKarzhZOFIk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solutions to Basic AND Compound Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LP3fsZNjJkc\">https:\/\/youtu.be\/LP3fsZNjJkc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Solve a Compound Inequality Involving OR (Union). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/oRlJ8G7trR8\">https:\/\/youtu.be\/oRlJ8G7trR8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Solve a Compound Inequality Involving AND (Intersection). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UU_KJI59_08\">https:\/\/youtu.be\/UU_KJI59_08<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><li>Ex 1: Solve and Graph Basic Absolute Value inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0cXxATY2S-k\">https:\/\/youtu.be\/0cXxATY2S-k<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: Solve and Graph Absolute Value inequalities . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 3: Solve and Graph Absolute Value inequalitie. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ttUaRf-GzpM\">https:\/\/youtu.be\/ttUaRf-GzpM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. Value). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5jRUuiMUxWQ\">https:\/\/youtu.be\/5jRUuiMUxWQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Cecilia Venn Diagram\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Internet Privacy\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solutions to Basic OR Compound Inequalities\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/nKarzhZOFIk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Solutions to Basic AND Compound Inequalities\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LP3fsZNjJkc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Solve a Compound Inequality Involving OR (Union)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/oRlJ8G7trR8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Solve a Compound Inequality Involving AND (Intersection)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/UU_KJI59_08\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"cc\",\"description\":\"Ex 1: Solve and Graph Basic Absolute Value inequalities\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/0cXxATY2S-k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Solve and Graph Absolute Value inequalities \",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 3: Solve and Graph Absolute Value inequalitie\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ttUaRf-GzpM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. 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