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Section4.10Linear Inequalities in Two Variables

We have learned how to graph lines like \(y=2x+1\text{.}\) In this section, we will learn how to graph linear inequalities like \(y\gt2x+1\text{.}\)

Figure4.10.1Alternative Video Lesson
Example4.10.2Office Supplies

Michael has a budget of \(\$133.00\) to purchase some staplers and markers for the office supply closet. Each stapler costs \(\$19.00\text{,}\) and each marker costs \(\$1.75\text{.}\) If we use variable names so that he will purchase \(x\) staplers and \(y\) markers. Write and plot a linear inequality to model the relationship between the number of staplers and markers Michael can purchase. Keep in mind that Michael imight not spend all of the \(\$133.00\text{.}\)

The cost of buying \(x\) staplers would be \(19x\) dollars. Similarly, the cost of buying \(y\) markers would be \(1.75y\) dollars. Since whatever Michael spends needs to be no more than \(133\) dollars, we have the inequality

\begin{equation*} 19x+1.75y\leq133\text{.} \end{equation*}

This is a standard-form inequality, similar to EquationĀ (4.7.1). Next, let's graph it.

The first method to graph the inequality is to graph the corresponding equation, \(19x+1.75y=133\text{.}\) Its \(x\)- and \(y\)-intercepts can be found this way:

\begin{align*} 19x+1.75y\amp=133 \amp 19x+1.75y\amp=133\\ 19x+1.75(\substitute{0})\amp=133 \amp 19(\substitute{0})+1.75y\amp=133\\ 19x\amp=133 \amp 1.75y\amp=133\\ \divideunder{19x}{19}\amp=\divideunder{133}{19} \amp \divideunder{1.75y}{1.75}\amp=\divideunder{133}{1.75}\\ x\amp=7 \amp y\amp=76 \end{align*}

So the intercepts are \((7,0)\) and \((0,76)\text{,}\) and we can plot the line in FigureĀ 4.10.3.

Figure4.10.3\(19x+1.75y=133\)

The points on this line represent ways in which Michael can spend exactly all of the \(\$133\text{.}\) But what does a point like \((2,40)\) in FigureĀ 4.10.4, which is not on the line, mean in this context? That would mean Michael bought \(2\) staplers and \(40\) markers, spending \(19\cdot2+1.75\cdot40=108\) dollars. That is within Michael's budget.

In fact, any point on the lower left side of this line represents a total purchase within Michael's budget. The shading in FigureĀ 4.10.5 captures all solutions to \(19x+1.75y\leq133\text{.}\) Some of those solutions have negative \(x\)- and \(y\)-values, which make no sense in context. So in FigureĀ 4.10.6, we restrict the shading to solutions which make physical sense.

Figure4.10.4The line \(19x+1.75y=133\) with a point identified that is within Michael's budget.
Figure4.10.5Shading all points that solve the inequality.
Figure4.10.6Shading restricted to points that make physical sense in context.

Let's look at some more examples of graphing linear inequalities in two variables.

Example4.10.7

Is the point \((1,2)\) a solution of \(y\gt2x+1\text{?}\)

In the inequality \(y\gt2x+1\text{,}\) substitute \(x\) with \(1\) and \(y\) with \(2\text{,}\) and we will see whether the inequality is true:

\begin{align*} y\amp\gt2x+1\\ 2\amp\stackrel{?}{\gt}2(2)+1\\ 2\amp\stackrel{\text{no}}{\gt}5 \end{align*}

Since \(2\gt5\) is not true, \((1,2)\) is not a solution of \(y\gt2x+1\text{.}\)

Example4.10.8

Graph \(y\gt2x+1\text{.}\)

There are two steps to graphing this linear inequality in two variables.

  1. Graph the line \(y=2x+1\text{.}\) Because the inequality symbol is \(\gt\) (instead of \(\ge\)), the line should be dashed (instead of solid).
  2. Next, we need to decide whether to shade the region above \(y=2x+1\) or below it. We will choose a point to test whether \(y\gt2x+1\) is true. As long as the line doesn't cross \((0,0)\text{,}\) we will use \((0,0)\) to test because the number \(0\) is the easiest number for calculation. \begin{align*} y\amp\gt2x+1\\ 0\amp\stackrel{?}{\gt}2(0)+1\\ 0\amp\stackrel{no}{\gt}1 \end{align*} Because \(0\gt1\) is not true, the point \((0,0)\) is not a solution and should not be shaded. As a result, we shade the region without \((0,0)\text{.}\)
A coordinate plane with y=2x+1 graphed as a dashed line.
A coordinate plane with y=2x+1 graphed as a dashed line and the region above the line is shaded.
Figure4.10.9Step 1 of graphing \(y\gt2x+1\)
Figure4.10.10Complete graph of \(y\gt2x+1\)
Example4.10.11

Graph \(y\leq -\frac{5}{3}x+2\text{.}\)

There are two steps to graphing this linear inequality in two variables.

  1. Graph the line \(y= -\frac{5}{3}x+2\text{.}\) Because the inequality symbol is \(\leq\) (instead of \(\lt\)), the line should be solid.
  2. Next, we need to decide whether to shade the region above \(y= -\frac{5}{3}x+2\) or below it. We will choose a point to test whether \(y\leq -\frac{5}{3}x+2\) is true there. Using \((0,0)\) as a test point: \begin{align*} y\amp\leq -\frac{5}{3}x+2\\ 0\amp\stackrel{?}{\leq}-\frac{5}{3}(0)+2\\ 0\amp\stackrel{\checkmark}{\leq}2 \end{align*} Because \(0\leq2\) is true, the point \((0,0)\) is a solution. As a result, we shade the region with \((0,0)\text{.}\)
A coordinate plane with y=-5/3x+2 graphed as a solid line.
A coordinate plane with y=-5/3x+2 graphed as a solid line and the region below the line is shaded.
Figure4.10.12Step 1 of graphing \(y\leq -\frac{5}{3}x+2\)
Figure4.10.13Complete graph of \(y\leq -\frac{5}{3}x+2\)

Subsection4.10.1Exercises

Exercises on Graphing Two-Variable Inequalities

1

Graph the linear inequality \(y\geq -4x\text{.}\)

2

Graph the linear inequality \(y\leq -\frac{1}{2}x-3\text{.}\)

3

Graph the linear inequality \(y\lt 3x+5\text{.}\)

4

Graph the linear inequality \(y\gt \frac{4}{3}x+1\text{.}\)

5

Graph the linear inequality \(2x+y\geq 3\text{.}\)

6

Graph the linear inequality \(3x+2y\lt -6\text{.}\)

7

Graph the linear inequality \(y\geq 3\text{.}\)

8

Graph the linear inequality \(x\lt-1\text{.}\)

Application Exercises on Graphing Two-Variable Inequalities

9

You fed your grandpa's cat while he was on vacation. When he was back, he took out a huge bank of coins, including quarters and dimes. He said you can take as many coins as you want, but the total value must be less than \(\$30.00\text{.}\)

  1. Write an inequality to model this situation, with \(q\) representing the number of quarters you will take, and \(d\) representing the number of dimes.

  2. Graph this linear inequality.

10

A couple is planning their wedding. They want the cost of the reception and the ceremony to be no more than \(\$8{,}000\text{.}\)

  1. Write an inequality to model this situation, with \(r\) as the cost of the reception (in dollars) and \(c\) as the cost of the ceremony (in dollars).

  2. Graph this linear inequality.