#### 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

- 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.
- 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?
- Referring to Q1 and Q2, how many hours does Amanda spend before taking the break?
- What does an absolute value make?
- Solve for x, if |x| + 5 = 7.
- Solve for x. |3x – 5| = 1
- Solve for x, 4| x + 5| + 3 > 7
- Solve for x, |x| – 4 < 6
- 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.
- Solve |x|< 7.

#### Concept Map:

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