###### 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

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:

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.