Absolute Value Equations and Inequalities

Introduction

In the previous session, we learned about compound inequality and solving compound inequalities.

Now we will learn about absolute equations and inequalities.

Bella went to a contest where she needed to select 10 items, the cost of each item should be between $9 to $100.

Which of the items shown she can select?

Write a model equation for this.

Will she win the contest?

Absolute Value Equation

The absolute value equation is the equation, where the variable is within the absolute value operator.

Example 1:

What is the value of x in |x| + 3 = 8

Solution:

Solve ‘x’ by isolating the absolute value on one side of the absolute expression.

|x| + 3 – 3 = 8 – 3

|x| = 5

The solutions are x = -5 and x = 5

Verifying solutions:

7 = |-5| + 2

= 5 + 2

= 7 ✓

7 = |5| + 2

= 5 + 2

= 7✓

Example2:

What is the value of x in |3x – 5 | = 4?

Solution:

There are two possibilities

Positive:

3x – 5 = 4

3x – 5 + 5 = 4 + 5

3x = 9

x = 3

Negative:

3x – 5 = -4

3x – 5 + 5 = -4 + 5

3x = 1

Note:

The expression inside the absolute value symbol can be positive or negative.

So 3x – 5 = -4 or 4

Example3:

What is the value of 5|x+1| + 9 = 4

Solution:

5 |x + 1| + 9 = 4

5|x + 1|+ 9 – 9 = 4 – 9

5 |x + 1| = -5

|x + 1| = -1

There is no solution.

Note:

The absolute value of a number is distance; hence it cannot be negative.

Real Life Example

Mr. A is participating in a cycling competition if his speed is 11mi/hr.  He plans to participate in this race within 5 miles or 39 miles. What equation models the number of hours, Mr. A will be in the race?

What will be the minimum and the maximum number of hours Mr. A will be in Race?

Solution:

Model equation:

|11x – 39| = 5

11x – 39 = 5               or 11x – 39 = -5

11x – 39 + 39 = 5 + 39                           

11x = 44

x = 4

Mr. A will be in the race for at least 3.36 hours and at most 4 hours

Absolute Value Inequality

An absolute value inequality is an expression that has an absolute function as well as an inequality sign.

What are the solutions to absolute value inequality?

Example:

Now we will solve and graph two absolute value inequalities.

|x|<5

-5 < x < 5

The solutions are -5 < x, x < 5, which are compound inequalities

The distance between x and 0 should be less than 5 units so, the values 5 units to the right and 5 units to the left is the solution

|x|>5

The solutions are x < -5, x > 5 which are compound inequalities

The distance between x and 0 is greater than 5 units. So, the positive values of x must be greater than 5 and the negative values of x must be less than -5.

Example1:

Solve |x|≤ 15.

Now we will solve and graph two absolute value inequalities.

|x| ≤ 15

-15 ≤ x ≤ 15

The solutions are -15 ≤ x, x ≤ 15, which are compound inequalities

The distance between x and 0 should be less than 15 units so, the values 15 units to the right and 15 units to the left are the solution.

Real Life Example

In a school, students are taken for an exhibition, which has some water activities as shown in the image. Each student must select 5 water activities. The total cost for each student must be within $ 45 of $500.

Solution:

The model equation is

|5x – 500| ≤ 45.

5x – 500 ≤ 45

5x – 500 + 500 ≤ 45 + 500

5x ≤ 545

x ≤ 109

The model equation is

|5x – 500| ≤ 45.

5x – 500 ≥ -45

5x – 500 + 500 ≥ -45 + 500

5x ≥ 455

x ≥ 91

The cost of the ride should be between $ 91 and $ 109 inclusively. So, the students can choose 4 among these 5 water activities as shown in the image

Exercise

1. Amanda is participating in a 50-mile spin thon. Her spin bike keeps track of the simulated number of miles she travels. She plans to take a 10-minute break within 5 miles of riding 25 miles. Write an absolute inequality that models the number of miles Amanda spins before taking a break.
2. Refer to Q1 and answer If Amanda spins at a constant speed of 12 miles/hr. How is the distance traveled by Amanda related to the time taken before taking the break?
3. Referring to Q1 and Q2, how many hours does Amanda spend before taking the break?
4. What does an absolute value make?
5. Solve for x, if |x| + 5 = 7.
6. Solve for x. |3x – 5| = 1
7. Solve for x, 4| x + 5| + 3 > 7
8. Solve for x, |x| – 4 < 6
9. The boat speed is 14 mi/hr. If the boat goes to a spot within 5 miles of 50 miles. Determine the number of hours the boat travels.
10. Solve |x|< 7.

Concept Map: