# Absolute value and inequality - Examples, Exercises and Solutions

An absolute value is the distance from the zero point,
that is, it does not refer to the sum of the number (whether negative or positive),
but it focuses on how far it is from the point $0$.

#### The absolute value is symbolized as follows :$││$

Generally we can write:
$│-X│= │X│= X$

## Practice Absolute value and inequality

### Exercise #1

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 #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

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 #4

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 #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-4\right|<8

Which of the following statements is necessarily true?

-4 < x < 12

### Exercise #2

Given:

\left|x+2\right|<3

Which of the following statements is necessarily true?

-5 < x < 1

### Exercise #3

Given:

\left|x+4\right|>13

Which of the following statements is necessarily true?

x>9 o x<-17

### Exercise #4

$\left|x-10\right|=0$

### Video Solution

$x=10$

### Exercise #5

$\left|6x-12\right|=6$

### Video Solution

$x=1$ , $x=3$

### Exercise #1

$\left|x+1\right|=5$

### Exercise #2

$\left|x\right|=5$

### Exercise #3

$\left|x-1\right|=6$

### Video Solution

$x=-5$ , $x=7$

### Exercise #4

Given:

\left|x-5\right|>11

Which of the following statements is necessarily true?

x>16 o x<-6

### Exercise #5

Given:

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

Which of the following statements is necessarily true?