Solving Systems of Linear Inequalities Graphically

To determine the system shown by the region, we can start with the boundary lines. The dashed line is vertical. If it were extended, we'd see that it passes through the $x$-axis at $x=6.$ Additionally, since the region to the right of $x=6$ is shaded, values of $x$ that are greater than $6$ are included. Since the line is dashed, $x=6$ is not included in the set. This means the inequality symbol is strict. Thus, $$x>6.$$ The solid line is not vertical, so we can write its equation in slope-intercept form. Let's start by identifying two points on the line.

Two points on the line are $(8,\text{-}18)$ and $(12,\text{-}24).$ We can use their coordinates to find the slope. The slope of the line is $m=\text{-}1.5.$ We can write the incomplete equation of the line as $$y=mx+b\Rightarrow y=\text{-}1.5x+b.$$ To find the $y$-intercept, we can substitute either point into the above equation for $x$ and $y.$ We'll choose $(8,\text{-}18).$ $$y=\text{-}1.5x+b\Rightarrow \text{-}18=\text{-}1.5\cdot 8+b.$$ Solving this equation gives $b.$ The $y$-intercept is $b=\text{-}6.$ Thus, the equation of the boundary line for the second inequality is $$y=\text{-}1.5x-6.$$ The shaded region is below the line means that the inequality sign is either $<$ or $\le .$ Since the line is solid, the inequality symbol is $\le .$ The system of inequalities that describes the shaded region is $$\{\begin{array}{l}x>6\\ y\le \text{-}1.5x-6.\end{array}$$ To verify that this is the correct system, let's graph the inequalities and see if we get the same region. The first inequality is $x$ greater than $6.$ That is all points to the right of $x=6,$ and since the inequality is strict, the line should be dashed.

The second inequality represent the region below the boundary line $y=\text{-}1.5x-6.$ The inequality isn't strict so the line should be solid.

The region that contains the solution set of both inequalities is shown by the overlapping red and blue shading.

The region is the same. Thus, the system of inequalities created is correct.

There are a number of conditions in this exercise. Let's start with making sense of them. Let's use $b$ to represent the number of burritos that Marco buys, and $t$ to represent the number of tacos. Since the burritos cost $\$5$ each, the total amount spent on just burriots is $5b.$ Similarly, since a taco costs $\$3,$ the total cost of tacos is $3t.$ Therefore, The total cost can be expressed as $$5b+3t.$$ Since Marco has $\$90,$ he cannot spend more than this. The total must be less than or equal to $90.$ That means, $$5b+3t\le 90.$$ We also know that Marco needs to buy enough food to feed $10$ people. Assuming no one wants to share a taco or a burrito, the total amount of dishes must be at least $10.$ This gives $$b+t\ge 10.$$ Since both of these inequalities must hold true, we get the following system of inequalities. $$\{\begin{array}{l}5b+3t\le 90\\ b+t\ge 10\end{array}$$ Solving this system gives all of the possible combinations of burritos and tacos Marco can purchase while buying enough food and staying within his budget. To solve the system, we must graph it. Let's first write the inequalities in slope-intercept form by isolating $b.$ The first inequality can be expressed as $b\le \text{-}0.6t+18.$ Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting $t$ on both sides. $$b+t\ge 10\iff b\ge \text{-}t+10$$ The system can now be expressed as $$\{\begin{array}{l}b\le \text{-}0.6t+18\\ b\ge \text{-}t+10.\end{array}$$ We'll graph each inequality by showing its boundary line and shading the appropriate region.

The purple region represents the solution set that satisfies both inequalities. Since Marco can't buy a negative number of burritos or tacos we're only interested in the positive values of $b$ and $t.$

Any point in this region corresponds to a combination on burritos and tacos that costs less than $\$90$ and feeds at least $10$ people. Let's look at the corners of this region.

The marked points represent minimum and maximum possibilities.

• $(10,0)$ and $(0,10)$ represent Marco buying either $10$ tacos or $10$ burritos. Then he'd feed exactly $10$ people, and have money remaining.

• $(0,18)$ and $(30,0)$ tell how many of each dish Marco can buy if he used all money on and only bought tacos or burritos. Thus, he can buy $18$ burritos or $30$ tacos.

Since we don't know how hungry the guests are, or what their preferences are, Marco should buy both burritos and tacos. Let's choose a point in the middle of the region.

One possibility is that Marco can purchase $12$ tacos and $8$ burritos. That way, there will probably be enough food, and he'll have money left. Note that even though decimal numbers are a part of the solution set, the answer should be given in whole numbers, assuming you can't buy a part of a taco or burrito.