Example 4.10.2 Office Supplies

Isabel 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 \(\$1.75\text{.}\) We will define the variables so that she will purchase \(x\) staplers and \(y\) markers. Write and plot a linear inequality to model the relationship between the number of staplers and markers Isabel can purchase. Keep in mind that she might 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 Isabel 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 Equation (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 Figure 4.10.3.

a Cartesian graph of a line with an x-intercept of (7,0) and a y-intercept of (0,76); the x-axis represents the number of staplers and the y-axis represents the number of markers
Figure 4.10.3 \(19x+1.75y=133\)

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

In fact, any point on the lower left side of this line represents a total purchase within Isabel's budget. The shading in Figure 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 Figure 4.10.6, we restrict the shading to solutions which make physical sense.

Figure 4.10.4 The line \(19x+1.75y=133\) with a point identified that is within Isabel's budget.
Figure 4.10.5 Shading all points that solve the inequality.
Figure 4.10.6 Shading restricted to points that make physical sense in context.