algebra 2 honors unit i: equations and inequalities 1.6- inequalities 1.7- absolute value equations

Algebra 2 HonorsUnit I: Equations and Inequalities

1.6- Inequalities1.7- Absolute Value Equations

1.6 Inequalities Math Vocabulary

EXAMPLE 1 Solve a simple absolute value equation

Solve |x – 5| = 7. Graph the solution.

SOLUTION

| x – 5 | = 7

x – 5 = – 7 or x – 5 = 7

Write original equation.

Write equivalent equations.

EXAMPLE 1

The solutions are –2 and 12. These are the values of x that are 7 units away from 5 on a number line. The graph is shown below.

ANSWER

Solve a simple absolute value equation

EXAMPLE 2 Solve an absolute value equation

Solve |5x – 10 | = 45.

EXAMPLE 2 Solve an absolute value equation

The solutions are 11 and –7. Check these in the original equation.

ANSWER

Check:| 5x – 10 | = 45 | 5x – 10 | = 45

EXAMPLE 3

Solve |2x + 12 | = 4x. Check for extraneous solutions.

Check for extraneous solutions

EXAMPLE 3

| 2x + 12 | = 4x

Check the apparent solutions to see if either is extraneous.

Check for extraneous solutions

| 2x + 12 | = 4x

CHECK

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

1. | x | = 5

for Examples 1, 2 and 3

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

4. |3x – 2| = 13

for Examples 1, 2 and 3

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

3. |2x + 5| = 3x

for Examples 1, 2 and 3

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

6. |4x – 1| = 2x + 9

for Examples 1, 2 and 3

ANSWER

The solutions are – and 5. 31 1

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

2. |x – 3| = 10

for Examples 1, 2 and 3

The solutions are –7 and 13. These are the values of x that are 10 units away from 3 on a number line. The graph is shown below.

ANSWER

– 3

– 4

– 2

– 1

0

1

2

3

4

5

6

7

– 5

– 6

– 7

8

9

10

11

12

13

10 10

GUIDED PRACTICE

Solve the equation. Check for extraneous solutions.

3. |x + 2| = 7

for Examples 1, 2 and 3

The solutions are –9 and 5. These are the values of x that are 7 units away from – 2 on a number line.

ANSWER

EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c

A professional baseball should weigh 5.125 ounces, with a tolerance of 0.125 ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball.

Baseball

SOLUTION

Write a verbal model. Then write an inequality.STEP 1

EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c

STEP 2 Solve the inequality.

Write inequality.

Write equivalent compound inequality.

Add 5.125 to each expression.

|w – 5.125| ≤ 0.125

– 0.125 ≤ w – 5.125 ≤ 0.125

5 ≤ w ≤ 5.25

So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below.

ANSWER

EXAMPLE 6

The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.25 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses.

Gymnastics

SOLUTION

STEP 1 Calculate the mean of the extreme mat thicknesses.

Write a range as an absolute value inequality

EXAMPLE 6

Mean of extremes = = 7.875 7.5 + 8.25 2

Find the tolerance by subtracting the mean from the upper extreme.

STEP 2

Tolerance = 8.25 – 7.875

Write a range as an absolute value inequality

= 0.375

EXAMPLE 6

STEP 3 Write a verbal model. Then write an inequality.

A mat is acceptable if its thickness t satisfies |t – 7.875| ≤ 0.375.

ANSWER

Write a range as an absolute value inequality

GUIDED PRACTICE for Examples 5 and 6

Solve the inequality. Then graph the solution.

10. |x + 2| < 6

The solutions are all real numbers less than – 8 or greater than 4. The graph is shown below.

ANSWER

–8 < x < 4

GUIDED PRACTICE for Examples 5 and 6

Solve the inequality. Then graph the solution.

11. |2x + 1| ≤ 9

The solutions are all real numbers less than –5 or greater than 4. The graph is shown below.

ANSWER

–5 ≤ x ≤ 4

GUIDED PRACTICE for Examples 5 and 6

12. |7 – x| ≤ 4

Solve the inequality. Then graph the solution.

3 ≤ x ≤ 11

ANSWER

The solutions are all real numbers less than 3 or greater than 11. The graph is shown below.

GUIDED PRACTICE for Examples 5 and 6

13. Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses.

A mat is unacceptable if its thickness t satisfies |t – 7.875| > 0.375.

ANSWER