{"id":6712,"date":"2020-10-08T15:10:49","date_gmt":"2020-10-08T15:10:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6712"},"modified":"2026-02-05T06:55:26","modified_gmt":"2026-02-05T06:55:26","slug":"2-1-linear-inequalities-with-one-variable","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/2-1-linear-inequalities-with-one-variable\/","title":{"raw":"2.1: Linear Inequalities with One Variable","rendered":"2.1: Linear Inequalities with One Variable"},"content":{"raw":"
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section 2.1 Learning Objectives<\/h3>\r\n2.1:\u00a0 Linear Inequalities with One Variable<\/strong>\r\n
    \r\n \t
  • Express solutions to inequalities graphically, with interval notation, and with set-builder notation<\/li>\r\n \t
  • Solve single-step linear inequalities with one variable<\/li>\r\n \t
  • Combine properties of inequality to solve linear inequalities with one variable<\/li>\r\n \t
  • Simplify using the Distributive Property and solve linear inequalities with one variable<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Writing inequalities<\/h2>\r\nFirst, let's define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right\u2014just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.\r\n\r\nThe first way you are probably familiar with\u2014the basic inequality. For example:\r\n
      \r\n \t
    • [latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than 9. Would you rather write\u00a0[latex]{x}\\lt{9}[\/latex] or try to list all<\/strong> the possible numbers that are less than 9? (hopefully, your answer is \"no\", you don't want to try to list them all.)<\/li>\r\n \t
    • [latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\r\n<\/ul>\r\nNote how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.\r\n\r\nFor example:\r\n
        \r\n \t
      • [latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\r\n \t
      • [latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.<\/li>\r\n<\/ul>\r\nThe second way is with a graph using the number line:\r\n

        \"Number<\/p>\r\nThe third way is with an interval.\r\n\r\nThe fourth way is using set-builder notation.\r\n\r\nWe will explore the second, third, and fourth ways in depth in this section. Again, those four ways to write solutions to inequalities are:\r\n

          \r\n \t
        • an inequality<\/li>\r\n \t
        • a graph<\/li>\r\n \t
        • an interval<\/li>\r\n \t
        • using set-builder notation<\/li>\r\n<\/ul>\r\n

          Inequality Signs<\/h3>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it's easy to get tangled up in inequalities, just remember to read them from left to right.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Symbol<\/th>\r\nWords<\/th>\r\nExample<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          [latex]\\neq [\/latex]<\/td>\r\nis not equal to<\/td>\r\n[latex]{2}\\neq{8}[\/latex], 2<\/i>\u00a0is<\/strong> not equal<\/b> to 8.<\/em><\/td>\r\n<\/tr>\r\n
          [latex]\\gt[\/latex]<\/td>\r\nis greater than<\/td>\r\n[latex]{5}\\gt{1}[\/latex], 5<\/i>\u00a0is greater than<\/strong>\u00a01<\/i><\/td>\r\n<\/tr>\r\n
          [latex]\\lt[\/latex]<\/td>\r\nis less than<\/td>\r\n[latex]{2}\\lt{11}[\/latex], 2\u00a0<\/i>is less than<\/b>\u00a011<\/i><\/td>\r\n<\/tr>\r\n
          [latex] \\geq [\/latex]<\/td>\r\nis greater than or equal to<\/td>\r\n[latex]{4}\\geq{ 4}[\/latex], 4\u00a0<\/i>is greater than or equal to<\/b>\u00a04<\/i><\/td>\r\n<\/tr>\r\n
          [latex]\\leq [\/latex]<\/td>\r\nis less than or equal to<\/td>\r\n[latex]{7}\\leq{9}[\/latex], 7<\/i>\u00a0is less than or equal to<\/b>\u00a09<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe inequality [latex]x>y[\/latex]\u00a0can also be written as [latex]{y}<{x}[\/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.\r\n

          Graphing an inequality<\/h2>\r\nInequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. \u00a0Graphs are a very helpful way to visualize information - especially when that information represents an infinite list of numbers!\r\n\r\n[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to -4.\r\n\r\n\"Number\r\n\r\n[latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.\r\n\r\n\"Number\r\n\r\nEach of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\geq\\right) [\/latex] or less than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\leq\\right) [\/latex]. The point is part of the solution. An open circle is used for greater than<\/i> (>) or less than<\/i> (<). The point is not <\/i>part of the solution.\r\n\r\nThe graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.\r\n
          \r\n

          Example 1<\/h3>\r\nGraph the\u00a0inequality [latex]x\\ge 4[\/latex].\r\n[reveal-answer q=\"797241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797241\"]\r\n\r\nWe can use a number line as shown. Because the values for x<\/em> include 4, we place a solid dot on the number line at 4.\r\n\r\nThen we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\r\n

          \"Number<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis video shows an example of how to draw the graph of an inequality.\r\nhttps:\/\/youtu.be\/-kiAeGbSe5c\r\n

          \r\n

          Example 2<\/h3>\r\nWrite an inequality describing all the real numbers on the number line that are less than 2, then draw the corresponding graph.\r\n[reveal-answer q=\"867890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867890\"]\r\n\r\nWe need to start from the left and work right, so we start from negative infinity and end at [latex]2[\/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[\/latex].\r\n\r\nInequality: [latex]x<2[\/latex]\r\n\r\nTo draw the graph, place an open dot on the number line first, then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows how to write an inequality mathematically when it is given in words. We will then graph it.\r\nhttps:\/\/youtu.be\/E_ZWNVNEvOg\r\n

          Represent inequalities using interval notation<\/h2>\r\nAnother commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called\u00a0interval notation.\u00a0<\/strong>With this convention, sets are built\u00a0with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\\geq 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This method is widely used and will be present in other math courses you may take.\r\n\r\nThe main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\" A few examples of an interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. To construct intervals, start 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. While not technically numbers, this ordering extends to positive and negative infinity as well.\r\n\r\nThe table below describes all the possible inequalities that can occur and how to write them using interval notation, where a<\/em> and b<\/em> are real numbers.\u00a0\u00a0Remember to read inequalities from left to right, just like text.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Inequality<\/th>\r\nWords<\/th>\r\nInterval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          [latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\nall real numbers between\u00a0a<\/em> and b<\/em>, not including a and b<\/td>\r\n[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{x}\\gt{a}[\/latex]<\/td>\r\nAll real numbers greater than a<\/em>, but not including a<\/em><\/td>\r\n[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{x}\\lt{b}[\/latex]<\/td>\r\nAll real numbers less than b<\/em>, but not including b<\/em><\/td>\r\n[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{x}\\ge{a}[\/latex]<\/td>\r\nAll real numbers greater than a<\/em>, including a<\/em><\/td>\r\n[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{x}\\le{b}[\/latex]<\/td>\r\nAll real numbers less than b<\/em>, including b<\/em><\/td>\r\n[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\r\nAll real numbers between a <\/em>and b<\/em>, including a<\/em><\/td>\r\n[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\r\nAll real numbers between a<\/em> and b<\/em>, including b<\/em><\/td>\r\n[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\r\nAll real numbers between a <\/em>and b<\/em>, including a <\/em>and b<\/em><\/td>\r\n[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\r\nAll real numbers less than a<\/em> or greater than b<\/em><\/td>\r\n[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex] (*Note: This\u00a0\u222a symbol will be discussed in a future section)<\/td>\r\n<\/tr>\r\n
          All real numbers<\/td>\r\nAll real numbers<\/td>\r\n[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
          \r\n

          Example 3<\/h3>\r\nDescribe the inequality [latex]x\\ge 4[\/latex] using interval notation.\r\n[reveal-answer q=\"817362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"817362\"]\r\n\r\nThe solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].\r\n\r\nNote the use of a bracket on the left because 4 is<\/strong> included in the solution set.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of using interval notation to describe an inequality.\r\nhttps:\/\/youtu.be\/BKhDzNKjVBc\r\n
          \r\n

          Example 4<\/h3>\r\nUse interval notation to indicate all real numbers less than or equal to [latex]-2[\/latex].\r\n[reveal-answer q=\"961990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961990\"]\r\n\r\nUse a parentheses on the left of [latex]-\\infty[\/latex] and bracket after [latex]-2[\/latex]: [latex]\\left(-\\infty,-2 \\right][\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers less than or equal to -2, which includes all real numbers from negative infinity to [latex]-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of translating words into an inequality and writing it in interval notation, as well as drawing the graph.\r\nhttps:\/\/youtu.be\/OYkQ-McI2qg\r\n
          \r\n

          Think About It 1<\/h3>\r\nIn the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and\u00a0asked to write the inequality and draw the graph.\r\n\r\nGiven [latex]\\left(-\\infty,10\\right)[\/latex], write the associated inequality and draw the graph.\r\n\r\nIn the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"15120\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"15120\"]\r\n\r\nWe will draw the graph first.\r\n\r\nThe interval reads \"all real numbers less\u00a0than 10,\" so we will start by placing an open\u00a0dot on 10 and drawing a line to the left\u00a0with an arrow indicating the solution continues to negative infinity.\r\n

          \"Number<\/p>\r\nTo write the inequality, we will use < since the parentheses indicate that 10 is not included. [latex]x<10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see examples of how to draw a graph given an inequality in interval notation.\r\n\r\nhttps:\/\/youtu.be\/lkhILNEPbfk\r\n\r\nAnd finally, one last video that shows how to write inequalities using a graph, with interval notation and as an inequality.\r\n\r\nhttps:\/\/youtu.be\/X0xrHKgbDT0\r\n

          Represent inequalities using set-builder notation<\/strong><\/h2>\r\nAnother method used for describing inequalities and solutions to inequalities is called set-builder notation.\u00a0 In this method, we show the conditions a variable must meet in order to be part of a set.\r\n\r\nIf we want to describe the set of all Real numbers that are greater than 0 using set-builder notation, it would look like the following:\r\n

          [latex]{\\color{red}\\{}{\\color{green}x}\\hspace{.01in}{\\mathbf{\\color{blue}|}}\\hspace{.01in}x>0{\\color{red}\\}}[\/latex]<\/span><\/span><\/p>\r\n

          The parentheses indicate that we have a set. The vertical line, |, is read \"such that\". Therefore, the example above would read as:<\/p>\r\n

          \"<\/span>The set of<\/span> all x<\/span> such that<\/span>\u00a0x is greater than 0.\"<\/p>\r\nLet\u2019s look at one more example.\u00a0 Let\u2019s use set-builder notation to describe the set of all real numbers that are between 4 and 10, including 10.\r\n

          [latex]{\\color{red}\\{}{\\color{green}x}\\hspace{.01in}{\\mathbf{\\color{blue}|}}\\hspace{.01in}{4}\\lt{x}\\le{10}{\\color{red}\\}}[\/latex]<\/span><\/span><\/p>\r\n

          The set of<\/span> all x<\/span> such that<\/span>\u00a0x is between 4 and 10, including 10.<\/p>\r\n\r\n

          \r\n

          Example 5<\/h3>\r\nDescribe the inequality [latex]1\u2264x<8[\/latex] using set-builder notation.\r\n\r\n[reveal-answer q=\"267083\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"267083\"]\r\n\r\n[latex]\\{{x|1\u2264x<8}\\}[\/latex]\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n
          \r\n

          Example 6<\/h3>\r\nUse set-builder notation to indicate all Real numbers less than -6 .\r\n\r\n[reveal-answer q=\"781995\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"781995\"]\r\n\r\n[latex]\\{{x|x<-6}\\}[\/latex]\r\n\r\nThis describes the set of all Real numbers less than -6.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

          Solving single-step inequalities<\/h2>\r\n

          Solve inequalities with addition and subtraction<\/h3>\r\nYou can solve most inequalities using inverse operations \u00a0as you did for solving equations. \u00a0This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.\r\n
          \r\n

          Addition and Subtraction Properties of Inequality<\/h3>\r\nIf [latex]a>b[\/latex],\u00a0<\/i>then [latex]a+c>b+c[\/latex].\r\n\r\nIf\u00a0[latex]a>b[\/latex], <\/i>then [latex]a\u2212c>b\u2212c[\/latex].\r\n\r\n<\/div>\r\nBecause inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.\r\n
          \r\n

          Example 7<\/h3>\r\nSolve for x.<\/i>\r\n

          [latex] {x}+3\\lt{5}[\/latex]<\/p>\r\n[reveal-answer q=\"952771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"952771\"]\r\n\r\nIt is helpful to think of this inequality as asking you to find all the values for x<\/em>, including negative numbers, such that when you add three you will get a number less than 5.\r\n

          [latex] \\displaystyle \\begin{array}{l}x+3<\\,\\,\\,\\,5\\\\\\underline{\\,\\,\\,\\,\\,-3\\,\\,\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\,2\\,\\,\\end{array}[\/latex]<\/p>\r\nIsolate the variable by subtracting 3 from both sides of the inequality.\r\n

          Answer<\/h4>\r\nInequality: \u00a0[latex]x<2[\/latex]\r\n\r\nInterval: \u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]\r\n\r\nGraph: \"Number[\/hidden-answer]\r\n\r\n<\/div>\r\n

          The line represents all<\/em> the numbers to which\u00a0you can add 3 and get a number that is less than 5. There's a lot of numbers that solve this inequality!<\/p>\r\nJust as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.\r\n\r\nThe example below shows how you could check that [latex]x<2[\/latex]\u00a0<\/i>is the solution to [latex]x+3<5[\/latex].<\/i>\r\n

          \r\n

          Example 8<\/h3>\r\nCheck that [latex]x<2[\/latex]\u00a0<\/i>is the solution to [latex]x+3<5[\/latex].\r\n\r\n[reveal-answer q=\"811564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811564\"]\r\n\r\nSubstitute the end point 2 into the related equation, [latex]x+3=5[\/latex].\r\n

          [latex]\\begin{array}{r}x+3=5 \\\\ 2+3=5 \\\\ 5=5\\end{array}[\/latex]<\/p>\r\nPick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)\r\n

          \"Number<\/p>\r\n

          [latex] \\displaystyle \\begin{array}{r}x+3<5 \\\\ 0+3<5 \\\\ 3<5\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n\r\n[latex]x<2[\/latex] is the solution to [latex]x+3<5[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!\r\n

          \r\n

          Example 9<\/h3>\r\nSolve for x<\/em>:\u00a0[latex]\\hspace{.05in}x-10\\leq-12[\/latex]\r\n[reveal-answer q=\"815894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815894\"]\r\n\r\nIsolate the variable by adding 10 to both sides of the inequality.\r\n

          [latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\\\\\underline{\\,\\,\\,+10\\,\\,\\,\\,\\,+10}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,-2\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]x\\leq-2[\/latex]\r\nInterval: [latex]\\left(-\\infty,-2\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d [latex]\\left(\\leq\\right)[\/latex]. The blue arrow is drawn to the left of the point [latex]\u22122[\/latex] because these are the values that are less than [latex]\u22122[\/latex].\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]x-10\\leq -12[\/latex]\r\n[reveal-answer q=\"268062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"268062\"]\r\n\r\nSubstitute the end point [latex]\u22122[\/latex] into the related equation \u00a0[latex]x-10=\u221212[\/latex]\r\n

          [latex] \\displaystyle \\begin{array}{r}x-10=-12\\,\\,\\,\\\\\\text{Does}\\,\\,\\,-2-10=-12?\\\\-12=-12\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22122[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

          [latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,-5-10\\le -12?\\\\-15\\le -12\\,\\,\\,\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\leq -2[\/latex]\u00a0is the solution to [latex]x-10\\leq -12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          \r\n

          Example 10<\/h3>\r\nSolve for a<\/em>:\u00a0[latex]\\hspace{.05in}a-17>-17[\/latex]\r\n[reveal-answer q=\"343031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343031\"]\r\n\r\nIsolate the variable by adding 17 to both sides of the inequality.\r\n

          [latex] \\displaystyle \\begin{array}{r}a-17>-17\\\\\\underline{\\,\\,\\,+17\\,\\,\\,\\,\\,+17}\\\\a\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,\\,\\,\\,\\,0\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality:\u00a0[latex] \\displaystyle a\\,\\,>\\,0[\/latex]\r\n\r\nInterval: [latex]\\left(0,\\infty\\right)[\/latex] \u00a0Note how we use parentheses on the left to show that the solution does not include 0.\r\n\r\nGraph: Note the open circle to show that the solution does not include 0.\r\n
          \r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution to\u00a0[latex]a-17>-17[\/latex]\r\n[reveal-answer q=\"653357\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653357\"]\r\n\r\nIs\u00a0[latex] \\displaystyle a\\,\\,>\\,0[\/latex] the correct solution to\u00a0\u00a0[latex]a-17>-17[\/latex]?\r\n\r\nSubstitute the end point 0 into the related equation.\r\n

          [latex] \\displaystyle \\begin{array}{r}a-17=-17\\,\\,\\,\\\\\\text{Does}\\,\\,\\,0-17=-17?\\\\-17=-17\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

          [latex] \\displaystyle \\begin{array}{r}a-17>-17\\,\\,\\,\\\\\\text{Is }\\,\\,20-17>-17?\\\\3>-17\\,\\,\\,\\\\\\\\\\text{It checks!}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex] \\displaystyle a\\,>\\,0[\/latex] is the solution to\u00a0[latex]a-17>-17[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nThe previous examples showed you how to solve a one-step inequality with the variable on the left hand side. \u00a0The following video provides examples of how to solve the same type of inequality.\r\n\r\nhttps:\/\/youtu.be\/1Z22Xh66VFM\r\n\r\nWhat would you do if the variable were on the right side of the inequality? \u00a0In the following example, you will see how to handle this scenario.\r\n

          \r\n

          Example 11<\/h3>\r\nSolve for x<\/em>:\u00a0[latex]\\hspace{.05in}4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"815893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815893\"]\r\n\r\nIsolate the variable by subtracting 5 from both sides of the inequality.\r\n

          [latex] \\displaystyle \\begin{array}{r}4\\geq{x}+5 \\\\\\underline{\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\,\\,\\,x\\end{array}[\/latex]<\/p>\r\n

          Rewrite the inequality with the variable on the left - this makes writing the interval and drawing the graph easier.<\/p>\r\n

          [latex]x\\le{-1}[\/latex]<\/p>\r\n

          Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]x\\le{-1}[\/latex] This can also be written as\r\nInterval: [latex]\\left(-\\infty,-1\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d . The blue arrow is drawn to the left of the point [latex]\u22121[\/latex] because these are the values that are less than [latex]\u22121[\/latex].\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"568062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568062\"]\r\n\r\nSubstitute the end point [latex]\u22121[\/latex] into the related equation \u00a0[latex]4=x+5[\/latex]\r\n

          [latex] \\displaystyle \\begin{array}{r}4=x+5\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4=-1+5?\\\\-1=-1\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22121[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n

          [latex] \\displaystyle \\begin{array}{r}4\\geq{-5}+5\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,4\\ge 0?\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\le{-1}[\/latex] is the solution to [latex]4\\geq{x}+5[\/latex][\/hidden-answer] <\/span>\r\n\r\n<\/div>\r\nThe following video show examples of solving inequalities with the variable on the right side.\r\nhttps:\/\/youtu.be\/RBonYKvTCLU\r\n

          Solve inequalities with multiplication and division<\/h3>\r\nSolving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let\u2019s look at what happens to the inequality when you multiply or divide each side by the same number.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          \r\n

          Analyzing an inequality, where both sides are multiplied by the same Positive<\/span>\u00a0number<\/strong><\/p>\r\n<\/td>\r\n

          		\r\n

          Analyzing an inequality, where both sides are multiplied by the same Negative\u00a0<\/span>number<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/thead>\r\n

          Let's start with the true statement:\r\n\r\n[latex]10>5[\/latex]<\/td>\r\nLet's try again by starting with the same true statement:\r\n\r\n[latex]10>5[\/latex]<\/td>\r\n<\/tr>\r\n
          Next, multiply both sides by the same positive number:\r\n\r\n[latex]10\\cdot 2>5\\cdot 2[\/latex]<\/td>\r\nThis time, multiply both sides by the same negative number:\r\n\r\n[latex]10\\cdot-2>5\\cdot -2[\/latex]<\/td>\r\n<\/tr>\r\n
          20 is greater than 10, so you still have a true inequality:\r\n\r\n[latex]20>10[\/latex]<\/td>\r\nWait a minute! [latex]\u221220[\/latex] is not <\/i>greater than [latex]\u221210[\/latex], so you have an untrue statement.\r\n\r\n[latex]\u221220>\u221210[\/latex]<\/td>\r\n<\/tr>\r\n
          When you multiply by a positive number, leave the inequality sign as it is!<\/td>\r\nYou must \u201creverse\u201d the inequality sign to make the statement true, whenever we multiply or divide by a negative number:<\/span>\r\n\r\n[latex]\u221220<\u221210[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
          \"Caution\"Remember! \u00a0When you multiply or divide by a negative number, \u201creverse\u201d the inequality sign. \u00a0 Whenever you multiply or divide both sides of an inequality by a negative\u00a0number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.<\/strong><\/div>\r\n
          \r\n

          Multiplication and Division Properties of Inequality<\/h3>\r\n\r\n\r\n\r\n\r\n\r\n
          Start With<\/strong><\/td>\r\nMultiply By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
          [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex], where\u00a0[latex]c[\/latex] is a positive number<\/td>\r\n[latex]ac>bc[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex], where\u00a0[latex]c[\/latex] is a negative number<\/td>\r\n[latex]ac<bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
          Start With<\/strong><\/td>\r\nDivide By<\/strong><\/td>\r\nFinal Inequality<\/strong><\/td>\r\n<\/tr>\r\n
          [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex], where\u00a0[latex]c[\/latex] is a positive number<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]a>b[\/latex]<\/td>\r\n[latex]c[\/latex], where\u00a0[latex]c[\/latex] is a negative number<\/td>\r\n[latex] \\displaystyle \\frac{a}{c}<\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n<\/div>\r\nKeep in mind that you only change the sign when you are multiplying and dividing by a negative<\/i> number. If you add or subtract<\/em> by a negative\u00a0number, the inequality stays the same.\r\n
          \r\n

          Example 12<\/h3>\r\nSolve for x:\u00a0<\/i>[latex]\\hspace{.05in}3x>12[\/latex]\r\n\r\n[reveal-answer q=\"691711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"691711\"]Divide both sides by 3 to isolate the variable.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\underline{3x}>\\underline{12}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\x>4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point 4, and then checking another solution for the inequality.\r\n

          [latex]\\begin{array}{r}3\\cdot4=12\\\\12=12\\\\3\\cdot10>12\\\\30>12\\\\\\text{It checks!}\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\n

          Inequality: [latex] \\displaystyle x>4[\/latex]<\/p>\r\n

          Interval: [latex]\\left(4,\\infty\\right)[\/latex]<\/p>\r\n

          Graph: \"Number<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThere was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive<\/i> 3. In the next example, there is division by a negative number, so there is an additional step in the solution!\r\n

          \r\n

          Example 13<\/h3>\r\nSolve for x<\/em>:\u00a0[latex]\\hspace{.05in}\u22122x>6[\/latex]\r\n\r\n[reveal-answer q=\"604033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604033\"]Divide each side of the inequality by [latex]\u22122[\/latex] to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\underline{-2x}>\\underline{\\,6\\,}\\\\-2\\,\\,\\,\\,-2\\,\\\\x<-3\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point [latex]\u22123[\/latex], and then checking another solution for the inequality.\r\n

          [latex]\\begin{array}{r}-2\\left(-3\\right)=6 \\\\6=6\\\\ -2\\left(-6\\right)>6 \\\\ 12>6\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n

          Answer<\/h4>\r\nInequality: [latex] \\displaystyle x<-3[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, -3\\right)[\/latex]\r\n\r\nGraph: \"Number\r\nBecause both sides of the inequality were divided by a negative number, [latex]\u22122[\/latex], the inequality symbol was switched from > to <.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of solving one step inequalities using the Multiplication Property of Equality where the variable is on the left hand side.\r\n\r\nhttps:\/\/youtu.be\/IajiD3R7U-0\r\n
          \r\n

          Think About It 2<\/h3>\r\nBefore you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.\r\n\r\nSolve for x<\/em>:\u00a0[latex]\\hspace{.05in}-\\frac{1}{2}>-12x[\/latex]\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"811465\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811465\"]\r\n\r\nThis inequality has the variable on the right hand side, which is different from the previous examples. Start the solution process as before, and at the end, you can move the variable to the left to write the final solution.\r\n\r\nDivide both sides by [latex]-12[\/latex] to isolate the variable. Since you are dividing by a negative number, you will need to change the direction of the inequality sign.\r\n

          [latex]\\displaystyle\\begin{array}{l}-\\frac{1}{2}\\gt{-12x}\\\\\\\\\\frac{-\\frac{1}{2}}{-12}\\gt\\frac{-12x}{-12}\\\\\\end{array}[\/latex]<\/p>\r\nNotice the change of direction of the inequality sign below. Also, dividing a fraction by an integer requires you to multiply by the reciprocal, and the reciprocal of [latex]-12[\/latex] is [latex]\\frac{1}{-12}[\/latex]\r\n

          [latex]\\displaystyle\\begin{array}{r}\\left(-\\frac{1}{12}\\right)\\left(-\\frac{1}{2}\\right)\\lt\\frac{-12x}{-12}\\,\\,\\\\\\\\ \\frac{1}{24}\\lt\\frac{\\cancel{-12}x}{\\cancel{-12}}\\\\\\\\ \\frac{1}{24}\\lt{x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\n

          Inequality: [latex]\\frac{1}{24}\\lt{x}[\/latex] \u00a0This can also be written with the variable on the left as [latex]x\\gt\\frac{1}{24}[\/latex]. \u00a0Writing the inequality with the variable on the left requires a little thinking, but helps you write the interval and draw the graph correctly.<\/p>\r\n

          Interval: [latex]\\left(\\frac{1}{24},\\infty\\right)[\/latex]<\/p>\r\n

          Graph:\u00a0\"Number<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives examples of how to solve an inequality with the Multiplication Property of Equality where the variable is on the right hand side.\r\n\r\nhttps:\/\/youtu.be\/s9fJOnVTHhs\r\n

          Combining properties of inequality to solve algebraic\u00a0inequalities<\/h2>\r\nA popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.\r\n
          \r\n

          Example 14<\/h3>\r\nSolve for p<\/i>:\u00a0[latex]\\hspace{.05in}4p+5<29[\/latex]\r\n\r\n[reveal-answer q=\"211828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211828\"]\r\n\r\nBegin to isolate the variable by subtracting 5 from both sides of the inequality.\r\n

          [latex] \\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by 4 to express the variable with a coefficient of 1.\r\n

          [latex]\\begin{array}{l}\\underline{4p}\\,<\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,<\\,\\,4\\\\\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality:\u00a0[latex]p<6[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\r\n\r\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where p<\/i> is less than 6 are found all along the number line to the left of 6.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"291597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"291597\"]\r\n\r\nCheck the end point 6 in the related equation.\r\n

          [latex] \\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nTry another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].\r\n

          [latex] \\displaystyle \\begin{array}{r}4p+5<29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5<29?\\\\0+5<29\\,\\,\\,\\\\5<29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]p<6[\/latex] is the solution to\u00a0[latex]4p+5<29[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          \r\n

          Example 15<\/h3>\r\nSolve for x<\/i>: \u00a0[latex]\\hspace{.05in}3x\u20137\\ge 41[\/latex]\r\n[reveal-answer q=\"238157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238157\"]\r\n\r\nBegin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.\r\n

          [latex] \\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]x\\ge 16[\/latex]\r\n\r\nInterval: [latex]\\left[16,\\infty\\right)[\/latex]\r\n\r\nGraph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"437341\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"437341\"]\r\n\r\nFirst, check the end point 16 in the related equation.\r\n

          [latex] \\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].\r\n

          [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.\r\n

          \r\n

          Example 16<\/h3>\r\nSolve for p<\/i>:\u00a0[latex]\\hspace{.05in}\u221258>14\u22126p[\/latex]\r\n\r\n[reveal-answer q=\"424351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"424351\"]\r\n\r\nNote how the variable is on the right hand side of the inequality, the method for solving does not change in this case.\r\n\r\nBegin to isolate the variable by subtracting 14 from both sides of the inequality.\r\n

          [latex] \\displaystyle \\begin{array}{l}\u221258\\,\\,>14\u22126p\\\\\\underline{-14\\,\\,\\,-14}\\\\-72\\,\\,>-6p\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.\r\n

          [latex]\\begin{array}{l}\\underline{-72}>\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\r\n

          We can also write this as [latex]p>12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]p>12[\/latex]\r\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]\r\nGraph: The graph of the inequality p <\/i>> 12 has an open circle at 12 with an arrow stretching to the right.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"500309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500309\"]\r\n\r\nFirst, check the end point 12 in the related equation.\r\n

          [latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Try 100.\r\n

          [latex]\\begin{array}{r}-58>14-6p\\\\-58>14-6\\left(100\\right)\\\\-58>14-600\\\\-58>-586\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the\u00a0left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/RB9wvIogoEM\r\n\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/9D2g_FaNBkY\r\n

          Simplifying and solving algebraic inequalities using the Distributive Property<\/h2>\r\nAs with equations, the Distributive Property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.\r\n
          \r\n

          Example 17<\/h3>\r\nSolve for x:<\/em>\u00a0[latex]\\hspace{.05in}2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]\r\n\r\n[reveal-answer q=\"587737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587737\"]\r\n\r\nDistribute to clear the parentheses.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\r\nSubtract 4x <\/i>from both sides to get the variable term on one side only.\r\n

          [latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides to isolate the variable.\r\n

          [latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides by 2 to express the variable with a coefficient of 1.\r\n

          [latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]x\\le8[\/latex]\r\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]\r\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"808701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808701\"]\r\n\r\nFirst, check the end point 8 in the related equation.\r\n

          [latex] \\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.\r\n

          [latex] \\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are given an example of how to solve a multi-step inequality that requires using the Distributive Property.\r\nhttps:\/\/youtu.be\/vjZ3rQFVkh8\r\n

          \r\n

          Think\u00a0About It 3<\/h3>\r\nIn the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.\r\n\r\nSolve for a: [latex]\\hspace{.05in}\\displaystyle\\frac{{2}{a}-{4}}{{-6}}{>-2}[\/latex]\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"701072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"701072\"]\r\n\r\nClear the fraction by multiplying both sides of the equation by [latex]-6[\/latex].\u00a0 While we encounter division much more often in the solving process, do not forget that multiplying by a negative number will also require us to reverse the inequality symbol.\r\n

          [latex]\\displaystyle \\begin{array}{r}\\frac{{2}{a}-{4}}{{-6}}{>-2}\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\-6\\,\\cdot \\,\\frac{2a-4}{-6}<-2\\,\\cdot \\,-6\\\\\\\\{2a-4}<12\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 4 to both sides to isolate the variable.\r\n

          [latex]\\displaystyle \\begin{array}{r}2a-4<12\\\\\\underline{\\,\\,\\,+4\\,\\,\\,\\,+4}\\\\2a<16\\end{array}[\/latex]<\/p>\r\nDivide both sides by 2 to express the variable with a coefficient of 1.<\/span>\r\n

          [latex] \\displaystyle \\begin{array}{c}\\frac{2a}{2}<\\,\\frac{16}{2}\\\\\\\\a<8\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]a<8[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,8\\right)[\/latex]\r\n\r\nGraph: The graph of this solution contains a solid dot at 8 to show that 8 is included in the solution set. The line continues to the left to show that values less than 8 are also included in the solution set.\"Number\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"905072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"905072\"]\r\nFirst, check the end point 8 in the related equation.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\frac{2a-4}{-6}=-2\\,\\,\\,\\,\\\\\\\\\\text{Does}\\,\\,\\,\\frac{2(8)-4}{-6}=-2?\\\\\\\\\\frac{16-4}{-6}=-2\\,\\,\\,\\,\\\\\\\\\\frac{12}{-6}=-2\\,\\,\\,\\,\\\\\\\\-2=-2\\,\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\text{Is}\\,\\,\\,\\frac{2(5)-4}{-6}>-2?\\\\\\\\\\frac{10-4}{-6}>-2\\,\\,\\,\\\\\\\\\\,\\,\\,\\,\\frac{6}{-6}>-2\\,\\,\\,\\\\\\\\-1>-2\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          <\/h2>\r\n

          Summary<\/h2>\r\nSolving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are four ways to represent solutions to inequalities: an inequality, a graph, an interval, and using set-builder notation.\u00a0Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality.\r\n\r\nInequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract either positive or negative<\/i> numbers to both sides of the inequality.\r\n\r\n \r\n

          <\/h2>","rendered":"
          \n

          section 2.1 Learning Objectives<\/h3>\n

          2.1:\u00a0 Linear Inequalities with One Variable<\/strong><\/p>\n

            \n
          • Express solutions to inequalities graphically, with interval notation, and with set-builder notation<\/li>\n
          • Solve single-step linear inequalities with one variable<\/li>\n
          • Combine properties of inequality to solve linear inequalities with one variable<\/li>\n
          • Simplify using the Distributive Property and solve linear inequalities with one variable<\/li>\n<\/ul>\n<\/div>\n

             <\/p>\n

            Writing inequalities<\/h2>\n

            First, let’s define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right\u2014just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.<\/p>\n

            The first way you are probably familiar with\u2014the basic inequality. For example:<\/p>\n

              \n
            • [latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than 9. Would you rather write\u00a0[latex]{x}\\lt{9}[\/latex] or try to list all<\/strong> the possible numbers that are less than 9? (hopefully, your answer is “no”, you don’t want to try to list them all.)<\/li>\n
            • [latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\n<\/ul>\n

              Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.<\/p>\n

              For example:<\/p>\n

                \n
              • [latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\n
              • [latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.<\/li>\n<\/ul>\n

                The second way is with a graph using the number line:<\/p>\n

                \"Number<\/p>\n

                The third way is with an interval.<\/p>\n

                The fourth way is using set-builder notation.<\/p>\n

                We will explore the second, third, and fourth ways in depth in this section. Again, those four ways to write solutions to inequalities are:<\/p>\n

                  \n
                • an inequality<\/li>\n
                • a graph<\/li>\n
                • an interval<\/li>\n
                • using set-builder notation<\/li>\n<\/ul>\n

                  Inequality Signs<\/h3>\n

                  The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it’s easy to get tangled up in inequalities, just remember to read them from left to right.<\/p>\n\n\n\n\n\n\n\n\n\n
                  Symbol<\/th>\nWords<\/th>\nExample<\/th>\n<\/tr>\n<\/thead>\n
                  [latex]\\neq[\/latex]<\/td>\nis not equal to<\/td>\n[latex]{2}\\neq{8}[\/latex], 2<\/i>\u00a0is<\/strong> not equal<\/b> to 8.<\/em><\/td>\n<\/tr>\n
                  [latex]\\gt[\/latex]<\/td>\nis greater than<\/td>\n[latex]{5}\\gt{1}[\/latex], 5<\/i>\u00a0is greater than<\/strong>\u00a01<\/i><\/td>\n<\/tr>\n
                  [latex]\\lt[\/latex]<\/td>\nis less than<\/td>\n[latex]{2}\\lt{11}[\/latex], 2\u00a0<\/i>is less than<\/b>\u00a011<\/i><\/td>\n<\/tr>\n
                  [latex]\\geq[\/latex]<\/td>\nis greater than or equal to<\/td>\n[latex]{4}\\geq{ 4}[\/latex], 4\u00a0<\/i>is greater than or equal to<\/b>\u00a04<\/i><\/td>\n<\/tr>\n
                  [latex]\\leq[\/latex]<\/td>\nis less than or equal to<\/td>\n[latex]{7}\\leq{9}[\/latex], 7<\/i>\u00a0is less than or equal to<\/b>\u00a09<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  The inequality [latex]x>y[\/latex]\u00a0can also be written as [latex]{y}<{x}[\/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.\n\n\n

                  Graphing an inequality<\/h2>\n

                  Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. \u00a0Graphs are a very helpful way to visualize information – especially when that information represents an infinite list of numbers!<\/p>\n

                  [latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to -4.<\/p>\n

                  \"Number<\/p>\n

                  [latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.<\/p>\n

                  \"Number<\/p>\n

                  Each of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\geq\\right)[\/latex] or less than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\leq\\right)[\/latex]. The point is part of the solution. An open circle is used for greater than<\/i> (>) or less than<\/i> (<). The point is not <\/i>part of the solution.<\/p>\n

                  The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex]\\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.<\/p>\n

                  \n

                  Example 1<\/h3>\n

                  Graph the\u00a0inequality [latex]x\\ge 4[\/latex].<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  We can use a number line as shown. Because the values for x<\/em> include 4, we place a solid dot on the number line at 4.<\/p>\n

                  Then we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.<\/p>\n

                  \"Number<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                  This video shows an example of how to draw the graph of an inequality.
                  \n