Linear Inequalities in Two Variables

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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)#.

  • 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:

    The direction of the inequality will tell you this information.

    Explanation:

    Example: #y >= 2x+3#

    You would draw the line #y = 2x+3# and shade above the line, since #y# is also greater than # 2x+3#.

    graph{y>=2x+3 [-10, 10, -5, 5]}

    Example: #y < 1/2x-2#

    You would draw the line #y=1/2x-2# as a dashed line, then shade below the line since #y# is less than #1/2x-2#.

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

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