# Absolute value inequality - Examples, Exercises and Solutions

## Inequality

When you come across signs like $<$ or ,$>$ you will know it is an inequality.
The result of the inequality will be a certain range of values that you will have to find.

An important rule to keep in mind: when you double or divide both sides of the operation, the sign of the inequality is reversed!

## Absolute Value Inequality

We can solve absolute value inequalities in $2$ ways:
In the geometric method and in the algebraic method.

## Practice Absolute value inequality

### Exercise #1

Solve the following inequality:

5x+8<9

### Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

x<\frac{1}{5}

### Exercise #2

Solve the inequality:

5-3x>-10

### Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

5 > x

### Exercise #3

Which diagram represents the solution to the inequality below? 5-8x<7x+3

### Step-by-Step Solution

First, we will move the elements:

5-8x>7x+3

5-3>7x+8x
2>13x

We divide the answer by 13, and we get:

x > \frac{2}{13}

### Exercise #4

What is the solution to the following inequality?

$10x-4≤-3x-8$

### Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

$10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

### Exercise #5

Solve the inequality:

8x+a < 3x-4

### Step-by-Step Solution

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!

x < -\frac{1}{5}a-\frac{4}{5}

### Exercise #1

Given:

\left|x+2\right|<3

Which of the following statements is necessarily true?

-5 < x < 1

### Exercise #2

Given:

\left|x-4\right|<8

Which of the following statements is necessarily true?

-4 < x < 12

### Exercise #3

Given:

\left|x+4\right|>13

Which of the following statements is necessarily true?

x>9 o x<-17

### Exercise #4

What is the solution to the inequality shown in the diagram?

### Video Solution

$3 ≤ x$

### Exercise #5

Which inequality is represented by the numerical axis below?

-7 < x ≤ 2

### Exercise #1

Given:

\left|x-5\right|>11

Which of the following statements is necessarily true?

x>16 o x<-6

### Exercise #2

Given:

\left|x-5\right|>-11

Which of the following statements is necessarily true?

No solution

### Exercise #3

When are the following inequalities satisfied?

3x+4<9

3 < x+5

### Video Solution

-2 < x < 1\frac{2}{3}

### Exercise #4

Find when the inequality is satisfied:

-3x+15<3x<4x+8

2.5 < x

### Exercise #5

Which diagram corresponds to the inequality below?

$40x+57≤5x-13≤25x+7$

What is its solution?