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Solving Systems of Linear Inequalities Graphically

A system of inequalities is a set of two or more inequalities analyzed together. In this course, linear systems in two inequalities will be explored. This section discusses how to graph and solve these systems.

System of Inequalities

The following system of inequalities contains two conditions on the variables xx and y.y. {y-0.5x+3y>x \begin{cases}y \leq \text{-} 0.5x+3 \\ y > x \end{cases} Systems of inequalities are often illustrated graphically in a coordinate plane, where the inequalities define a region.


Graphing a System of Linear Inequalities

Individual inequalities can be interpreted graphically as the area above or below a boundary line. When all inequalitites in a system are graphed, the solution to the system is the overlapping region of the individual solution sets. For example, the following system can be solved in this way. {x+y<7x+2y<10 \begin{cases}x+y<7 \\ x+2y<10 \end{cases}


Write the inequalities in slope-intercept form
To be able to graph the inequalities individually, start by writing them in slope-intercept form.
{x+y<7(I)x+2y<10(II)\begin{cases}x+y<7 & \, \text {(I)}\\ x+2y<10 & \text {(II)}\end{cases}
{y<-x+72y<-x+10\begin{cases}y<\text{-} x + 7 \\ 2y<\text{-} x + 10 \end{cases}
{y<-x+7y<-0.5x+5\begin{cases}y<\text{-} x + 7 \\ y<\text{-} 0.5x + 5 \end{cases}


Graph the inequalities in a coordinate plane

To graph the inequalities, begin with the boundary lines. The inequality y<-x+7y<\text{-} x + 7 has the boundary line y=-x+7.y=\text{-} x + 7. Since the inequality sign is <,<, the line should be dashed, and the region below the line is shaded.

Similarly, y<-0.5x+5y<\text{-} 0.5 x + 5 has the boundary line y=-0.5x+5.y=\text{-}0.5x+5. The inequality sign is <,<, so the line is dashed and the region below the line is shaded.


Find the overlapping region

Notice that the individual solution sets overlap for a portion of the graph. This overlapping region is the solution set of the system. The points in this region are all the points that satisfy both inequalities. In this case, this is the purple region.

Lastly, since the boundary lines in their entirety are not part of the solution set, trim them to only show the borders of the overlapping region.


Write a system of inequalities that describes the shaded region.

Show Solution

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 xx-axis at x=6.x=6. Additionally, since the region to the right of x=6x=6 is shaded, values of xx that are greater than 66 are included. Since the line is dashed, x=6x=6 is not included in the set. This means the inequality symbol is strict. Thus, x>6. 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,-18)(8,\text{-}18) and (12,-24).(12,\text{-}24). We can use their coordinates to find the slope.
m=y2y1x2x1m = \dfrac{y_2-y_1}{x_2-x_1}
Evaluate right-hand side
m=-24(-18)128m = \dfrac{{\color{#0000FF}{\text{-}24}}-\left({\color{#009600}{\text{-}18}} \right)}{{\color{#0000FF}{12}}-{\color{#009600}{8}}}
The slope of the line is m=-1.5.m={\color{#0000FF}{\text{-}1.5}}. We can write the incomplete equation of the line as y=mx+by=-1.5x+b. y = mx+ b \quad \Rightarrow \quad y={\color{#0000FF}{\text{-} 1.5}}x+b. To find the yy-intercept, we can substitute either point into the above equation for xx and y.y. We'll choose (8,-18).({\color{#0000FF}{8}},{\color{#009600}{\text{-} 18}}). y=-1.5x+b-18=-1.58+b. y=\text{-} 1.5x+b \quad \Rightarrow \quad {\color{#009600}{\text{-} 18}}=\text{-} 1.5 \cdot {\color{#0000FF}{8}}+b. Solving this equation gives b.b.
-18=-1.58+b\text{-}18=\text{-}1.5 \cdot 8+b
Solve for bb
The yy-intercept is b=-6.b=\text{-}6. Thus, the equation of the boundary line for the second inequality is y=-1.5x6. y=\text{-}1.5x-6. The shaded region is below the line means that the inequality sign is either << or .\leq. Since the line is solid, the inequality symbol is .\leq. The system of inequalities that describes the shaded region is {x>6y-1.5x6. \begin{cases}x>6 \\ y\leq \text{-}1.5x-6. \end{cases} 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 xx greater than 6.6. That is all points to the right of x=6,x=6, and since the inequality is strict, the line should be dashed.

The second inequality represent the region below the boundary line y=-1.5x6.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.


Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him $90 \$90 and instructs him to get enough food so that they can feed 1010 people. If burritos cost $5 \$5 each and tacos cost $3 \$ 3 each, how many of each can he buy?

Show Solution
There are a number of conditions in this exercise. Let's start with making sense of them. Let's use bb to represent the number of burritos that Marco buys, and tt to represent the number of tacos. Since the burritos cost $5\$ 5 each, the total amount spent on just burriots is 5b.5b. Similarly, since a taco costs $3,\$ 3, the total cost of tacos is 3t.3t. Therefore, The total cost can be expressed as 5b+3t. 5b+3t. Since Marco has $90,\$ 90, he cannot spend more than this. The total must be less than or equal to 90.90. That means, 5b+3t90. 5b+3t\leq 90. We also know that Marco needs to buy enough food to feed 1010 people. Assuming no one wants to share a taco or a burrito, the total amount of dishes must be at least 10.10. This gives b+t10. b+t\geq 10. Since both of these inequalities must hold true, we get the following system of inequalities. {5b+3t90b+t10 \begin{cases}5b+3t\leq 90 \\ b+t\geq 10 \end{cases} 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.b.
5b+3t905b+3t\leq 90
Solve for bb
5b-3t+905b\leq \text{-} 3t+90
b-3t5+905b\leq \text{-} \dfrac{3t}{5} + \dfrac{90}{5}
b-0.6t+18b\leq \text{-} 0.6t+18
The first inequality can be expressed as b-0.6t+18.b\leq \text{-} 0.6t+18. Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting tt on both sides. b+t10b-t+10 b+t \geq 10 \quad \Leftrightarrow \quad b \geq \text{-} t+10 The system can now be expressed as {b-0.6t+18b-t+10.\begin{cases}b\leq \text{-} 0.6t+18 \\ b \geq \text{-} t+10. \end{cases} 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 bb and t.t.

Any point in this region corresponds to a combination on burritos and tacos that costs less than $90\$ 90 and feeds at least 1010 people. Let's look at the corners of this region.

The marked points represent minimum and maximum possibilities.

  • (10,0)(10,0) and (0,10)(0,10) represent Marco buying either 1010 tacos or 1010 burritos. Then he'd feed exactly 1010 people, and have money remaining.
  • (0,18)(0,18) and (30,0)(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 1818 burritos or 3030 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 1212 tacos and 88 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.

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