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Linear Systems

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

A system of inequalities is a set of two or more inequalities involving the same variables. For example, consider the set formed by the following two inequalities.
The solution set of a system of inequalities is the set of all ordered points that satisfy all the inequalities in the system simultaneously. Usually, systems of inequalities are solved by graphing each inequality on the same coordinate plane. By doing so, the entire coordinate plane is divided into different regions. By moving P, explore each region formed by the previous system.
Two lines graphed and four regions
Of the regions formed when a system of inequalities is graphed, the overlapping region represents the solution set of the system.


Solving a System of Linear Inequalities Graphically

A system of linear inequalities can be solved graphically by graphing all inequalities on the same coordinate plane and then finding the region of intersection, if any. For example, consider the following system.
To solve the previous system graphically, these three steps can be followed.


Write the Inequalities in Slope-Intercept Form
To be able to graph the inequalities individually, start by writing them in slope-intercept form.


Graph the Inequalities

To graph the inequalities, begin with the boundary lines. The inequality y<-x+7 has the boundary line y=-x+7. Since the inequality is strict, the line should be dashed and, in this case, the shaded region lies below the line.

Similarly, y<-0.5x+5 has the boundary line y=-0.5x+5. Since the inequality is not strict, the line is solid. In addition, the region to be shaded lies below the line. This inequality will be graphed on the same coordinate plane.


Find the Overlapping Region

Notice that the individual solution sets overlap in a portion of the plane. This overlapping region is the solution set of the system. All the points in this region satisfy both inequalities simultaneously. In the next graph, only the common region is plotted.

Finally, since the boundary lines in their entirety are not part of the solution set, crop them only to show the edges of the overlapping region.

Keep in mind that if there is no overlapping region, the system has no solution.

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 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,
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) and (12,-24). We can use their coordinates to find the slope.
Evaluate right-hand side
The slope of the line is We can write the incomplete equation of the line as
To find the y-intercept, we can substitute either point into the above equation for x and y. We'll choose
Solving this equation gives b.
Solve for b
The y-intercept is b=-6. Thus, the equation of the boundary line for the second inequality is
The shaded region is below the line means that the inequality sign is either < or . Since the line is solid, the inequality symbol is . The system of inequalities that describes the shaded region is
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=-1.5x6. 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 and instructs him to get enough food so that they can feed 10 people. If burritos cost $5 each and tacos cost $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 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
Since Marco has $90, he cannot spend more than this. The total must be less than or equal to 90. That means,
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
Since both of these inequalities must hold true, we get the following system of inequalities.
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.
Solve for b
The first inequality can be expressed as b-0.6t+18. Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting t on both sides.
The system can now be expressed as
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.

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