{"id":6654,"date":"2020-10-03T16:22:29","date_gmt":"2020-10-03T16:22:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6654"},"modified":"2026-01-17T22:36:03","modified_gmt":"2026-01-17T22:36:03","slug":"3-6-graphing-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-6-graphing-linear-inequalities\/","title":{"raw":"3.6: Graphing Linear Inequalities","rendered":"3.6: Graphing Linear Inequalities"},"content":{"raw":"
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section 3.6 Learning Objectives<\/h3>\r\n3.6: Graphing Linear Inequalities<\/strong>\r\n
    \r\n \t
  • Determine if an ordered pair satisfies a linear inequality<\/li>\r\n \t
  • Graph horizontal and vertical inequalities<\/li>\r\n \t
  • Graph a linear inequality in two variables<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Identify the difference between the graph of a linear equation and linear inequality<\/h2>\r\nRecall that solutions to linear inequalities typically lead to an infinite number of solutions represented by an interval of values.\r\n\r\nHere is an example from the section on solving linear inequalities:\r\n\r\nSolve for p<\/i>. [latex]4p+5<29[\/latex]\r\n

    [latex] \\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\underline{4p}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\underline{24}\\,\\,\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\r\nYou can interpret the solution as \"p<\/em> can be any number less than six.\"\r\n\r\nIn this chapter we have learned how to graph linear equations in two variables.\u00a0 For example, we previously showed how to graph the line described by this equation:\u00a0 [latex]y=2x+3[\/latex].\u00a0\u00a0We<\/strong><\/b>\u00a0found that we can construct a never-ending table of values that corresponds to points on the line\u2014these are some of the solutions to the equation [latex]y=2x+3[\/latex].\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/em> values<\/th>\r\n[latex]2x+3[\/latex]<\/th>\r\ny<\/em> values<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    0<\/td>\r\n[latex]2(0)+3[\/latex]<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
    1<\/td>\r\n[latex]2(1)+3[\/latex]<\/td>\r\n5<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n[latex]2(2)+3[\/latex]<\/td>\r\n7<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n[latex]2(3)+3[\/latex]<\/td>\r\n9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAdditionally, we learned how to graph the line that represents all the points that make\u00a0[latex]y=2x+3[\/latex] a true statement.\r\n\r\n\"Increasing\r\n\r\nWhat if we combined these two ideas\u2014linear inequalities and graphs of lines? First translate the line, [latex]y=2x+3[\/latex], into words:\r\n\r\nYou get y<\/em> by multiplying x<\/em> by two and adding three. [latex]y=2x+3[\/latex]\r\n\r\nHow would you translate this inequality into words? [latex]y<2x+3[\/latex]\r\n\r\nFor what values of x<\/em> will you get an output, y, that is less than<\/em> 2 times x<\/em> plus three?\r\n\r\nWOW, that may\u00a0seem confusing, but keep reading, we'll help you figure it out.\r\n\r\nLinear inequalities are different than linear equations, although you can apply what you know about equations to help you understand inequalities. Inequalities and equations are both math statements that compare two values. Equations use the symbol = ; recall that inequalities are\u00a0represented by the symbols < , \u2264 , > , and \u2265.\r\n\r\nOne way to visualize two-variable inequalities is to plot them on a coordinate plane. Here is what the inequality\u00a0[latex]x>y[\/latex]\u00a0<\/i>looks like. The solution is a region, which is shaded. This region is made up of lots and lots of ordered pairs that all make the statement\u00a0[latex]x>y[\/latex] true.\r\n\r\n\"Dotted\r\n\r\n \r\n\r\nThere are a few things to notice here. First, look at the dashed red boundary line: this is the graph of the related linear equation\u00a0[latex]x=y[\/latex]. Next, look at the light red region that is to the right of the line. This region (excluding the line [latex]x=y[\/latex]) represents the entire set of solutions for the inequality [latex]x>y[\/latex]. Remember how all points on a line<\/em> are solutions to the linear equation of the line? Well, all points in a region<\/em> are solutions to the linear inequality<\/b> representing that region.\r\n\r\nLet\u2019s think about it for a moment\u2014if [latex]x>y[\/latex], then a graph of [latex]x>y[\/latex]\u00a0will show all ordered pairs [latex](x,y)[\/latex] for which the x-<\/i>coordinate is greater than the y-<\/i>coordinate.\r\n\r\nThe graph below shows the region [latex]x>y[\/latex]\u00a0as well as some ordered pairs on the coordinate plane. Look at each ordered pair. Is the x-<\/i>coordinate greater than the y-<\/i>coordinate? Does the ordered pair sit inside or outside of the shaded region?\r\n\r\n\"Dotted\r\n\r\n \r\n\r\nThe ordered pairs [latex](4,0)[\/latex] and [latex](0,\u22123)[\/latex] lie inside the shaded region. In these ordered pairs, the x-<\/i>coordinate is larger than the y-<\/i>coordinate. These ordered pairs are in the solution set of the equation [latex]x>y[\/latex].\r\n\r\nThe ordered pairs [latex](\u22123,3)[\/latex] and [latex](2,3)[\/latex] are outside of the shaded area. In these ordered pairs, the x-<\/i>coordinate is smaller<\/i> than the y-<\/i>coordinate, so they are not included in the set of solutions for the inequality.\r\n\r\nThe ordered pair [latex](\u22122,\u22122)[\/latex] is on the boundary line. It is not a solution as [latex]\u22122[\/latex] is not greater than [latex]\u22122[\/latex]. However, had the inequality been [latex]x\\geq y[\/latex]\u00a0(read as \u201cx<\/i> is greater than or equal to y<\/i>\u201d), then [latex](\u22122,\u22122)[\/latex] would have been included (and the line would have been represented by a solid line, not a dashed line).\r\n\r\nhttps:\/\/youtu.be\/EcrLbRJ2zV0\r\n

    Determine if an ordered pair satisfies a linear inequality<\/h2>\r\nLet\u2019s take a look at another example: the inequality [latex]3x+2y\\leq6[\/latex]. The graph below shows the region of values that makes this inequality true (shaded red), the boundary line [latex]3x+2y=6[\/latex], as well as a handful of ordered pairs. The boundary line is solid this time, because points on the boundary line [latex]3x+2y=6[\/latex]\u00a0will make the inequality [latex]3x+2y\\leq6[\/latex]\u00a0true.\r\n\r\n\"A\r\n\r\n \r\n\r\nAs you did with the previous example, you can substitute the x-<\/i> and y-<\/i>values in each of the [latex](x,y)[\/latex] ordered pairs into the inequality to find solutions. While you may have been able to do this in your head for the inequality\u00a0[latex]x>y[\/latex], sometimes making a table of values makes sense for more complicated inequalities.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Ordered Pair<\/th>\r\nMakes the inequality\r\n\r\n[latex]3x+2y\\leq6[\/latex]\r\n\r\na true statement<\/th>\r\nMakes the inequality\r\n\r\n[latex]3x+2y\\leq6[\/latex]\r\n\r\na false statement<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex](\u22125, 5)[\/latex]<\/td>\r\n[latex]\\begin{array}{r}3\\left(\u22125\\right)+2\\left(5\\right)\\leq6\\\\\u221215+10\\leq6\\\\\u22125\\leq6\\end{array}[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex](\u22122,\u22122)[\/latex]<\/td>\r\n[latex]\\begin{array}{r}3\\left(\u22122\\right)+2\\left(\u20132\\right)\\leq6\\\\\u22126+\\left(\u22124\\right)\\leq6\\\\\u201310\\leq6\\end{array}[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex](2,3)[\/latex]<\/td>\r\n<\/td>\r\n[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(3\\right)\\leq6\\\\6+6\\leq6\\\\12\\leq6\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex](2,0)[\/latex]<\/td>\r\n[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(0\\right)\\leq6\\\\6+0\\leq6\\\\6\\leq6\\end{array}[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex](4,\u22121)[\/latex]<\/td>\r\n<\/td>\r\n[latex]\\begin{array}{r}3\\left(4\\right)+2\\left(\u22121\\right)\\leq6\\\\12+\\left(\u22122\\right)\\leq6\\\\10\\leq6\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf substituting [latex](x,y)[\/latex] into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region, or the point will be part of a dotted boundary line.\r\n\r\nhttps:\/\/youtu.be\/-x-zt_yM0RM\r\n
    \r\n

    Example 1<\/h3>\r\nUse the graph to determine which ordered pairs plotted below are solutions of the inequality\u00a0[latex]x\u2013y<3[\/latex].\r\n\r\n\"Upward-sloping\r\n\r\n[reveal-answer q=\"840389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840389\"]\r\n\r\nSolutions will be located in the shaded region. Since this is a \u201cless than\u201d problem, ordered pairs on the boundary line are not included in the solution set.\r\n\r\nThese values are located in the shaded region, so are solutions. (When substituted into the inequality\u00a0[latex]x\u2013y<3[\/latex], they produce true statements.)\r\n

    [latex](\u22121,1)[\/latex]<\/p>\r\n

    [latex](\u22122,\u22122)[\/latex]<\/p>\r\nThese values are not located in the shaded region, so are not solutions. (When substituted into the inequality [latex]x-y<3[\/latex], they produce false statements.)\r\n

    [latex](1,\u22122)[\/latex]<\/p>\r\n

    [latex](3,\u22122)[\/latex]<\/p>\r\n

    [latex](4,0)[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\n[latex](\u22121,1)\\,\\,\\,(\u22122,\u22122)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n

    Use a graph to determine ordered pair solutions of a linear inequality<\/h3>\r\nhttps:\/\/youtu.be\/GQVdDRVq5_o\r\n
    \r\n

    Example 2<\/h3>\r\nIs [latex](2,\u22123)[\/latex] a solution of the inequality [latex]y<\u22123x+1[\/latex]?\r\n\r\n[reveal-answer q=\"746731\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746731\"]\r\n\r\nIf [latex](2,\u22123)[\/latex] is a solution, then it will yield a true statement when substituted into the inequality\u00a0[latex]y<\u22123x+1[\/latex].\r\n

    [latex]y<\u22123x+1[\/latex]<\/p>\r\nSubstitute\u00a0[latex]x=2[\/latex] and [latex]y=\u22123[\/latex]\u00a0into inequality.\r\n

    [latex]\u22123<\u22123\\left(2\\right)+1[\/latex]<\/p>\r\nEvaluate.\r\n

    [latex]\\begin{array}{l}\u22123<\u22126+1\\\\\u22123<\u22125\\end{array}[\/latex]<\/p>\r\nThis statement is not <\/b>true, so the ordered pair [latex](2,\u22123)[\/latex] is not <\/b>a solution.\r\n

    Answer<\/h4>\r\n[latex](2,\u22123)[\/latex] is not a solution.[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/-x-zt_yM0RM\r\n

    Graph horizontal and vertical inequalities<\/h2>\r\nSometimes, you will be asked to graph an inequality that has only one variable. As we learned in graphing linear equations, these boundary lines will be either vertical or horizontal. To determine which side to shade, instead of choosing ordered pairs to test, we can simply shade the side that fits the description in the inequality.\r\n
    \r\n

    Example 3<\/h3>\r\nGraph the inequality [latex]y>2[\/latex].\r\n\r\n[reveal-answer q=\"137811\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"137811\"]\r\n\r\nThe boundary line for this inequality is the horizontal line\u00a0Graph the inequality [latex]y=2[\/latex]. We will make sure to make it a dotted line since this inequality symbol is strictly \"greater than\" and not \"greater than or equal to\". Since the inequality is asking us to make a graph representing all y-values greater than 2, we will shade above<\/em> the boundary line, where y-values are larger than 2.\r\n\r\n\"horizontal\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    Think about it<\/h3>\r\nWhat do you think the graph below should\u00a0look like? When trying the problem on your own below, make sure you are careful about grabbing the correct boundary line tool. (There is one that is dotted and one that is solid).\r\n\r\n[ohm_question hide_question_numbers=1]193768[\/ohm_question]\r\n\r\n<\/div>\r\n

    Graph a linear inequality in two variables<\/h2>\r\nSo how do you get from the algebraic form of an inequality, like [latex]y>3x+1[\/latex], to a graph of that inequality? Plotting inequalities is fairly straightforward if you follow a few steps.\r\n
    \r\n

    Graphing Inequalities<\/h3>\r\nTo graph an inequality:\r\n
      \r\n \t
    • Graph the related boundary line. Replace the <, >, \u2264 or \u2265 sign in the inequality with = to find the equation of the boundary line.<\/li>\r\n \t
    • Identify at least one ordered pair on either side of the boundary line and substitute those [latex](x,y)[\/latex] values into the inequality. Shade the region that contains the ordered pairs that make the inequality a true statement.\u00a0<\/b><\/li>\r\n \t
    • If points on the boundary line are solutions, then use a solid line for drawing the boundary line. This will happen for \u2264 or \u2265 inequalities.<\/li>\r\n \t
    • If points on the boundary line aren\u2019t solutions, then use a dotted line for the boundary line. This will happen for < or > inequalities.<\/li>\r\n<\/ul>\r\n<\/div>\r\nLet\u2019s graph the inequality [latex]x+4y\\leq4[\/latex].\r\n\r\nTo graph the boundary line, find at least two values that lie on the line [latex]x+4y=4[\/latex]. You can use the x<\/i>- and y<\/i>-intercepts for this equation by substituting 0 in for x<\/i> first and finding the value of y<\/i>; then substitute 0 in for y<\/i> and find x<\/i>.\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/i><\/b><\/td>\r\ny<\/i><\/b><\/td>\r\n<\/tr>\r\n
      0<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points [latex](0,1)[\/latex] and [latex](4,0)[\/latex], and draw a line through these two points for the boundary line. The line is solid because \u2264 means \u201cless than or equal to,\u201d so all ordered pairs along the line are included in the solution set.\r\n\r\n\"Solid\r\n\r\nThe next step is to find the region that contains the solutions. Is it above or below the boundary line? To identify the region where the inequality holds true, you can test a couple of ordered pairs, one on each side of the boundary line.\r\n\r\nIf you substitute [latex](\u22121,3)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:\r\n

      [latex]\\begin{array}{r}\u22121+4\\left(3\\right)\\leq4\\\\\u22121+12\\leq4\\\\11\\leq4\\end{array}[\/latex]<\/p>\r\nThis is a false statement, since 11 is not less than or equal to 4.\r\n\r\nOn the other hand, if you substitute [latex](2,0)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:\r\n

      [latex]\\begin{array}{r}2+4\\left(0\\right)\\leq4\\\\2+0\\leq4\\\\2\\leq4\\end{array}[\/latex]<\/p>\r\nThis is true! The region that includes [latex](2,0)[\/latex] should be shaded, as this is the region of solutions.\r\n\r\n\"Solid\r\n\r\nAnd there you have it\u2014the graph of the set of solutions for [latex]x+4y\\leq4[\/latex].\r\n\r\nhttps:\/\/youtu.be\/2VgFg2ztspI\r\n

      \r\n

      Example 4<\/h3>\r\nGraph the inequality [latex]2y>4x\u20136[\/latex].\r\n\r\n[reveal-answer q=\"138506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138506\"]\r\n\r\nSolve for y<\/i>.\r\n

      [latex] \\displaystyle \\begin{array}{r}2y>4x-6\\\\\\\\\\frac{2y}{2}>\\frac{4x}{2}-\\frac{6}{2}\\\\\\\\y>2x-3\\\\\\end{array}[\/latex]<\/p>\r\nCreate a table of values to find two points on the line [latex] \\displaystyle y=2x-3[\/latex], or graph it based on the slope-intercept method, the b<\/i> value of the y<\/i>-intercept is [latex]-3[\/latex] and the slope is 2.\r\n\r\nPlot the points, and graph the line. The line is dotted because the sign in the inequality is >, not \u2265 and therefore points on the line are not solutions to the inequality.\r\n\r\n\"Dotted\r\n

      [latex] \\displaystyle y=2x-3[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/th>\r\ny<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      0<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFind an ordered pair on either side of the boundary line. Insert the x<\/i>- and y<\/i>-values into the inequality\r\n[latex]2y>4x\u20136[\/latex] and see which ordered pair results in a true statement. Since [latex](\u22123,1)[\/latex] results in a true statement, the region that includes [latex](\u22123,1)[\/latex] should be shaded.\r\n

      [latex]\\begin{array}{l}2y>4x\u20136\\\\\\\\\\text{Test }1:\\left(\u22123,1\\right)\\\\2\\left(1\\right)>4\\left(\u22123\\right)\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2>\u201312\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2>\u221218\\\\\\text{TRUE}\\\\\\\\\\text{Test }2:\\left(4,1\\right)\\\\2(1)>4\\left(4\\right)\u2013 6\\\\\\,\\,\\,\\,\\,\\,2>16\u20136\\\\\\,\\,\\,\\,\\,\\,2>10\\\\\\text{FALSE}\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\nThe graph of the inequality [latex]2y>4x\u20136[\/latex] is:\r\n\r\n\"The\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA quick note about the problem above\u2014notice that you can use the points [latex](0,\u22123)[\/latex] and [latex](2,1)[\/latex] to graph the boundary line, but that these points are not included in the region of solutions, since the region does not include the boundary line!\r\n\r\nhttps:\/\/youtu.be\/Hzxc4HASygU\r\n

      Summary<\/h2>\r\nWhen inequalities are graphed on a coordinate plane, the solutions are located in a region of the coordinate plane, which is represented as a shaded area on the plane. The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of \u2264 and \u2265. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of < and >. You can tell which region to shade by testing some points in the inequality. Using a coordinate plane is especially helpful for visualizing the region of solutions for inequalities with two variables.","rendered":"
      \n

      section 3.6 Learning Objectives<\/h3>\n

      3.6: Graphing Linear Inequalities<\/strong><\/p>\n

        \n
      • Determine if an ordered pair satisfies a linear inequality<\/li>\n
      • Graph horizontal and vertical inequalities<\/li>\n
      • Graph a linear inequality in two variables<\/li>\n<\/ul>\n<\/div>\n

         <\/p>\n

        Identify the difference between the graph of a linear equation and linear inequality<\/h2>\n

        Recall that solutions to linear inequalities typically lead to an infinite number of solutions represented by an interval of values.<\/p>\n

        Here is an example from the section on solving linear inequalities:<\/p>\n

        Solve for p<\/i>. [latex]4p+5<29[\/latex]\n\n\n

        [latex]\\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\underline{4p}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\underline{24}\\,\\,\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\n

        You can interpret the solution as “p<\/em> can be any number less than six.”<\/p>\n

        In this chapter we have learned how to graph linear equations in two variables.\u00a0 For example, we previously showed how to graph the line described by this equation:\u00a0 [latex]y=2x+3[\/latex].\u00a0\u00a0We<\/strong><\/b>\u00a0found that we can construct a never-ending table of values that corresponds to points on the line\u2014these are some of the solutions to the equation [latex]y=2x+3[\/latex].<\/p>\n\n\n\n\n\n\n\n\n
        x<\/em> values<\/th>\n[latex]2x+3[\/latex]<\/th>\ny<\/em> values<\/th>\n<\/tr>\n<\/thead>\n
        0<\/td>\n[latex]2(0)+3[\/latex]<\/td>\n3<\/td>\n<\/tr>\n
        1<\/td>\n[latex]2(1)+3[\/latex]<\/td>\n5<\/td>\n<\/tr>\n
        2<\/td>\n[latex]2(2)+3[\/latex]<\/td>\n7<\/td>\n<\/tr>\n
        3<\/td>\n[latex]2(3)+3[\/latex]<\/td>\n9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Additionally, we learned how to graph the line that represents all the points that make\u00a0[latex]y=2x+3[\/latex] a true statement.<\/p>\n

        \"Increasing<\/p>\n

        What if we combined these two ideas\u2014linear inequalities and graphs of lines? First translate the line, [latex]y=2x+3[\/latex], into words:<\/p>\n

        You get y<\/em> by multiplying x<\/em> by two and adding three. [latex]y=2x+3[\/latex]<\/p>\n

        How would you translate this inequality into words? [latex]y<2x+3[\/latex]\n\nFor what values of x<\/em> will you get an output, y, that is less than<\/em> 2 times x<\/em> plus three?<\/p>\n

        WOW, that may\u00a0seem confusing, but keep reading, we’ll help you figure it out.<\/p>\n

        Linear inequalities are different than linear equations, although you can apply what you know about equations to help you understand inequalities. Inequalities and equations are both math statements that compare two values. Equations use the symbol = ; recall that inequalities are\u00a0represented by the symbols < , \u2264 , > , and \u2265.<\/p>\n

        One way to visualize two-variable inequalities is to plot them on a coordinate plane. Here is what the inequality\u00a0[latex]x>y[\/latex]\u00a0<\/i>looks like. The solution is a region, which is shaded. This region is made up of lots and lots of ordered pairs that all make the statement\u00a0[latex]x>y[\/latex] true.<\/p>\n

        \"Dotted<\/p>\n

         <\/p>\n

        There are a few things to notice here. First, look at the dashed red boundary line: this is the graph of the related linear equation\u00a0[latex]x=y[\/latex]. Next, look at the light red region that is to the right of the line. This region (excluding the line [latex]x=y[\/latex]) represents the entire set of solutions for the inequality [latex]x>y[\/latex]. Remember how all points on a line<\/em> are solutions to the linear equation of the line? Well, all points in a region<\/em> are solutions to the linear inequality<\/b> representing that region.<\/p>\n

        Let\u2019s think about it for a moment\u2014if [latex]x>y[\/latex], then a graph of [latex]x>y[\/latex]\u00a0will show all ordered pairs [latex](x,y)[\/latex] for which the x-<\/i>coordinate is greater than the y-<\/i>coordinate.<\/p>\n

        The graph below shows the region [latex]x>y[\/latex]\u00a0as well as some ordered pairs on the coordinate plane. Look at each ordered pair. Is the x-<\/i>coordinate greater than the y-<\/i>coordinate? Does the ordered pair sit inside or outside of the shaded region?<\/p>\n

        \"Dotted<\/p>\n

         <\/p>\n

        The ordered pairs [latex](4,0)[\/latex] and [latex](0,\u22123)[\/latex] lie inside the shaded region. In these ordered pairs, the x-<\/i>coordinate is larger than the y-<\/i>coordinate. These ordered pairs are in the solution set of the equation [latex]x>y[\/latex].<\/p>\n

        The ordered pairs [latex](\u22123,3)[\/latex] and [latex](2,3)[\/latex] are outside of the shaded area. In these ordered pairs, the x-<\/i>coordinate is smaller<\/i> than the y-<\/i>coordinate, so they are not included in the set of solutions for the inequality.<\/p>\n

        The ordered pair [latex](\u22122,\u22122)[\/latex] is on the boundary line. It is not a solution as [latex]\u22122[\/latex] is not greater than [latex]\u22122[\/latex]. However, had the inequality been [latex]x\\geq y[\/latex]\u00a0(read as \u201cx<\/i> is greater than or equal to y<\/i>\u201d), then [latex](\u22122,\u22122)[\/latex] would have been included (and the line would have been represented by a solid line, not a dashed line).<\/p>\n