ALI - Solving Systems of Linear Inequalities (Lesson)

Solving Systems of Linear Inequalities

In the real world, there are often times when more than one restriction or constraint might be involved. What if you were looking to order no less than 30 favors for your party but you also wanted to spend no more than $200? In this scenario, you would need to consider more than one inequality or a system of inequalities. We can determine the answer to a system of inequalities by graphing them on the same grid. 

Solving Systems of Inequalities Using the Graphing Method

Example: Graph the possible solutions for the following system of inequalities:

  • y < 2x + 1
  • y > 6x - 8 

First, let's graph these inequalities one at a time.

  • For y < 2x + 1, start at the y-intercept (0, 1) and move up and over according to the slope (up 2, right 1). 
  • We will use a solid line and shade to the right. 

2_5DesmosGraph1.png

Now, on the same graph, let's graph the other inequality.

  • For y > 6x - 8, start at the y-intercept (0, -8) and move up 6 and to the right 1. 
  • We will use a dotted line and shade to the left. 

2_5DesmosGraph2.png

The solution to our system of inequalities is where the shading of the graph overlaps. Let's test out our solution by choosing a coordinate point from the portion where the shadings overlap and substituting it into both inequalities. Let's choose the point (0, 0).

Inequality One

  • y < 2x + 1
  • 0 < 2(0) + 1
  • 0 < 1
  • This is true!

Inequality Two

  • y > 6x - 8
  • 0 > 6(0) - 8
  • 0 > -8
  • This is true!

Real World Systems of Inequalities

Example: Tati has two jobs. She works as a cashier, which pays $14 per hour and she works as a social media manager, which pays $25 per hour. She wants to earn at least $500 per week. Write an inequality to represent the situation.

14x + 25y > 500

Tati wants to work at most 30 hours a week. Write another inequality to represent the number of hours.

x + y < 30

Graph both inequalities. 

2_5DesmosGraph3.jpg

Since this is a real world scenario, we also have to determine an appropriate domain and range. The x-axis represents the number of hours worked as a cashier and the y-axis represents the number of hours worked as a social media manager. Since these both represent hours, the domain and the range cannot be less than zero. Since Tati also wants to work at most 30 hours, the domain and range must also be less than or equal to 30. So both the domain and range is [0, 30].

2_5DesmosGraph4.jpg

To check your solution, pick a point somewhere in the area where the shading overlaps. Let's choose (6, 20) and substitute it into the two equations.

Inequality One

  • 14x + 25y > 500
  • 14(6) + 25(20) > 500
  • 84 + 500 > 500
  • 584 > 500
  • This is true!

Inequality Two

  • x + y < 30
  • 6 + 20 < 30
  • 26 < 30
  • This is true!

 

Solving Systems of Linear Inequalities Practice

1. Graph the possible solutions for the following system of inequalities:

  • y > x + 4
  • y < 3x - 2

2. A delivery truck only carries 45 pound and 90 pound packages. For each delivery trip, the van must carry at least 15 packages, and the total weight of the packages cannot exceed 2000 pounds.

a. Write two inequalities that represent the situation.

b. Graph the possible solutions for this scenario.

c. Determine one possible solution using coordinates.

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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