Algebra 2 by Richard Wright

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0-07 Graph Linear Inequalities

Mr. Wright teaches the lesson.

Objectives:

SDA NAD Content Standards (2018): AII.4.1, AII.4.2, AII.5.2, AII.5.3, AII.7.1

tickets
Figure 1: Tickets (Pixabay/igorovsyannykov)

The junior class is selling tickets to their fundraiser. There are two types of tickets available: student and adult. The student tickets cost $8 each and the adult tickets cost $10 each. How many tickets need to be sold for the class revenue to be more than $2000? This can be written as 8s + 10a > 2000. Since there are two variables and an inequality, there are many solutions. A graph can be used to describe all the solutions.

Linear Inequality in Two Variables

Linear inequalities in two variables are like linear equations, but with inequality instead of an equals sign. One side of the inequality is larger than the other. Solutions are ordered pairs that give a true statement when substituted into the inequality.

Example 1

Tell whether the given ordered pair is a solution of 2x − 3y ≤ 8.

  1. (2, 0)
  2. (1, −4)

Solution

Substitute the ordered pair for x and y in the inequality and simplify. If the resulting statement is true, the ordered pair is a solution.

  1. 2(2) − 3(0) ≤ 8

    4 ≤ 8

    This is a true statement so (2, 0) is a solution.

  2. 2(1) − 3(−4) ≤ 8

    14 ≤ 8

    This is a false statement so (1, −4) is not a solution.

Graphing a Linear Inequality

A graph of a linear inequality looks like a line with have the coordinate plane shaded. The shaded area contains all the points that are solutions to the inequality.

linear inequality
Figure 2: Graph of a linear inequality.
Graph a Linear Inequality

To graph a linear inequality in two variables,

  1. Graph the line as if it was an equation.
  2. Decided is it is a dotted or solid line.
    1. The line is dotted if the inequality is not equal (<, >).
    2. The line is solid if the inequality is equal (≤, =, ≥).
  3. Shade the portion of the coordinate plane containing the solutions.
    1. Pick a test point not on the line. (0, 0) works well if the graphed line does not go through it.
    2. Substitute that point into the inequality to see if it is a solution.
    3. If the point is a solution, shade that side of the line containing that point.
    4. If the point is not a solution, shade the other side of the line.

Example 2

Graph x ≥ −4.

Solution

  1. Graph the line as if it were an equation. x = −4 is a vertical line.
  2. The line is solid because it is equal to, ≥.

  3. Pick (0, 0) as a test point. Substitute it into the inequality.

    0 ≥ −4

    This is a true statement so shade the side of the line with (0, 0).

x ≥ −4
Figure 3: x ≥ −4.

Example 3

Graph \(y < \frac{1}{3} x\).

Solution

  1. Graph the line as if it were an equation. The slope is \(\frac{1}{3}\) and the y-intercept is 0. So start at 0 on the y-axis and then go up 1 and over 3 several times to get more points.
  2. The line is dotted because it is not equal, <.

  3. Pick (1, 0) as a test point. (0, 0) will not work because the line goes through it. Substitute the test point into the inequality.

    $$ 0 < \frac{1}{3} \left(1\right) $$

    $$ 0 < \frac{1}{3} $$

    This a true statement so shade the side of the line with (1, 0).

y < 1/3 x
Figure 4: \(y < \frac{1}{3} x\).

Example 4

Graph y ≥ x – 2

Solution

The slope is 1 and the y-intercept is –2. Start at –2 on the y-axis and then go up 1 and over 1 several times to get more points.

The line is solid because it is equal, ≥.

Pick (0, 0) as a test point. Substitute the test point into the inequality.

0 ≥ 0 – 2

0 ≥ –2

This is a true statement so shade the side of the line with (0, 0).

y≥x-2
Figure 5: y ≥ x – 2

Example 5

Graph y > –|x – 1| + 2.

Solution

This is an absolute value graph. The vertex is (1, 2) and the slope of the right side is –1.

The line is dotted because it is not equal to, >.

Pick (0, 0) as a test point.

0 > –|0 – 1| + 2

0 > –1 + 2

0 > 1

This is a false statement so shade the other side of the line.

y>-|x-1|+2
Figure 6: y > –|x – 1| + 2

Example 6

You have two part-time summer jobs, one that pays $9 per hour and another that pays $10 per hour. You would like to earn at least $180 a week. a) Write an inequality describing the possible amounts of time you can schedule at both jobs. b) Graph the inequality. c) Identify three possible solutions of the inequality.

Solution

  1. Use the d = rt type formula where d = income, r = pay rate, and t = time at the job.

    d = r1t1 + r2t2

    180 ≤ 9t1 + 10t2

  2. This linear inequality is in standard form so find intercepts.

    $$ t_1 = \frac{180}{9} = 20 $$

    $$ t_2 = \frac{180}{10} = 18 $$

    Plot the intercepts and draw a line.

    The line is solid because it is equal, ≤.

    Pick (0, 0) as a test point.

    180 ≤ 9(0) + 10(0)

    180 ≤ 0

    This is false, so shade the other side of the line.

    180 ≤ 9t_1 + 10t_2
    Figure 7: 180 ≤ 9t1 + 10t2
  3. Some possible solutions are (12, 12), (10, 12), (6, 14), however, any points in the shaded area are solutions.

Practice Exercises

    Graph the linear inequality.

  1. y ≤ 3
  2. x > 2
  3. y ≤ 2x + 1
  4. \(y ≥\frac{2}{3}x-2\)
  5. y < –x + 3
  6. \(y<-\frac{1}{2}x+\frac{3}{2}\)
  7. 2x + 3y ≥ 6

    8. Graph the absolute value inequality.

  8. y > |x|
  9. y ≤ |x – 1| + 2
  10. \(y < -\frac{1}{2}\left|x\right|+3\)
  11. y ≥ 2|x – 1|

    12. Solve the real life problems.

  12. Joe is buying his sister some books and toy cars for her birthday. The books cost $3 each and cars cost $1 each, and Joe only has up to $20 to spend. a) Write an inequality describing the possible number of books and cars Joe can buy. b) Graph the inequality. c) Identify three possible solutions of the inequality.
  13. Frank the mechanic can only work up to 8 hours per day. Changing oil takes 0.5 hours and changing tires takes 1.5 hours. a) Write an inequality describing the possible number of oil changes and tire changes Frank can do in a day. b) Graph the inequality. c) Identify three possible solutions of the inequality.

    14. Mixed Review

  14. (0-06) Graph \(y=\frac{1}{2}\left|x+1\right|\)
  15. (0-05) Graph y = 4x − 1
  16. (0-05) Graph 3x − 6y = 12
  17. (0-04) Find the slope of the line that passes through (−3, 5) and (4, 6).
  18. (0-04) Write the equation of the line that passes through (−3, 5) and (4, 6).
  19. (0-04) Write the equation of the line that passes through (0, 3) and (2, 5).
  20. (0-03) Solve \(0=\frac{1}{2}\left|x+1\right|\)

Answers

  12. 3x + y ≤ 20; ; Sample: 3 books and 8 cars, 5 books and 1 car, 1 book and 14 cars
  13. 0.5x + 1.5y ≤ 8; ; Sample: 11 oil changes and 1 tire, 2 oil changes and 4 tires, 7 oil changes and 3 tires
  17. \(\frac{1}{7}\)
  18. \(y=\frac{1}{7}x+\frac{38}{7}\)
  19. y = x + 3
  20. −1