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 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 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 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
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 Figure 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 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.

the previous graph of the line 19x+1.75y=133 with the point (2,40) added; the point is below the line indicating it is within his budget
the previous graph of the line 19x+1.75y=133 with all of the points below the line shaded; these are solutions to the inequality
the previous graph of the line line 19x+1.75y=133 with only the points in the first quadrant that are below the line shaded
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.
in-context