Linear Inequalities in Two Variables

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5.5 - Lesson - Graphing Linear Inequalities Video Lesson
13:13 — by MrJSaintGermain

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Key Questions

  • The best way (when possible!) is to express the inequality bringing all the terms involing a variable on the left, and all the terms involving the other variable on the right. In the end, you'll have an inequality of the form #y\le f(x)# (or #y\ge f(x)#), and this is easy to graph, because if you can draw the graph of #f(x)#, then you'll have that #y\le f(x)# represents all the area under the function #f#, and #y\ge f(x)#, of course, the area over the function.

    For example, consider the inequality
    #yx^2<-y+3x#
    In this form, it would be very hard to say which points satisfy the inequality, but with some manipulations we obtain
    #yx^2+y<3x#
    #y(x^2+1)<3x#
    #y<\frac{3x}{x^2+1}#

    Now, the graph of #\frac{3x}{x^2+1}# is easy to draw, and the inequality is solved considering all the area below the graph, as showed:

    graph{y<3x/{x^2+1} [-10, 10, -5, 5]}

    Note that if you have #y<f(x)# the graph of the function #f# is not included, since it represents the points for which #y=f(x)#.

  • You check by using a point in your range.
    Take one point in the range (the shaded area) and check it again with your given equation.

    EX

    #y>2x+4# and #(1,7)# was your chose coordinate

    #7>2+4#
    #7>6#

    That is true, so your graph is correct.

    If you still doubt yourself, you can always check more points.

  • The solution set of a single linear inequality is always a half-plane, so there are infinitely many solutions.


    I hope that this was helpful.

  • Answer:

    See below.

    Explanation:

    Shading for absolute inequalities means shading over the solution region. The easiest way to determine shading for linear inequalities in two variables is to choose a test point and plug into the original inequality to determine if one should shade above or below the line.

    Steps for Shading Inequalities:

    1. Imagine that the inequality sign is an equal sign and graph the equation on a coordinate plane.
    2. If the inequality symbol is a #<=# or #>=#, then draw a solid line connecting the points. If the inequality symbol is a #<# or #>#, then draw a dashed line connecting the points.
    3. Choose a test point above the line and one that is below the line. Plug each ordered pair into the original inequality and one should be a false statement and one should be a true statement.
    4. Shade the region with the true statement.

    For example: Graph and shade #y >= 3x+5#

    Graph: #y=3x+5#

    enter image source here

    Choose a test point above the line #(-5,5)# and plug into the original inequality.

    #5>=3(-5)+5#

    #5>=-15+5#

    #5>=-10 -># True Statement

    Choose a test point below the line #(0,0)# and plug into the original inequality.

    #0>=3(0)+5#

    #0>=0+5#

    #0>=5 -># False Statement

    So we want to shade above the line:

    enter image source here

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