###### Example4.10.2Office 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.

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.

in-context