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# Absolute Value Equation

## Absolute value equations have two possible solutions. The equation |x| = 4. This means that x could be 4 or x could be -4. When you take the absolute value of 4, the solution is 4 and when you take the absolute value of -4, the solution is also 4. An absolute value problem, you have to get into account that there can be two solutions that will make the equation true. Learning absolute value equation, you set the quantity inside the absolute value symbol equal to the positive and negative value on the other side of the equal symbol. On a number line, the distance of a given number from zero gives the absolute value of that number. The direction on which it exists from zero, that is, left or right, does not matter in absolute value.

The absolutely value |x| can be defined as

|x| = $\left\{\begin{matrix} x; & x>0\\ -x;&x <0 \end{matrix}\right.$

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## Absolute Value Equations and Inequalities

To solve an absolute value equation, isolate the absolute value on one side of the equal sign.

### Solved Examples

Question 1: Solve |3x - 5| = 10
Solution:
Case 1: 3x - 5 = 10

$\Rightarrow$ 3x = 15

$\Rightarrow$ x = 5

Case 2: 3x - 5 = -10

$\Rightarrow$ 3x = - 5

$\Rightarrow$ x = $\frac{-5}{3}$.

The solution of this equation is 5, $\frac{-5}{3}$.

Question 2: Solve |2x - 3| $\leq$ 11
Solution:
Given, |2x - 3| $\leq$ 11

$\Rightarrow$ -11 $\leq$ 2x - 3 $\leq$ 11

$\Rightarrow$ -11 + 3 $\leq$ 2x - 3 + 3 $\leq$ 11 + 3

$\Rightarrow$ -8 $\leq$ 2x $\leq$ 14

$\Rightarrow$ -4 $\leq$ x $\leq$ 7

The solution of this inequality is [-4, 7].

## Graphing Absolute Value Equations

Absolute value equation have two possible solutions. There must be two different setups for each equation. Lets see with the help of example how to create the graph of the absolute equations.

### Solved Example

Question: Graph the following equations, y = |x| and x = 2.
Solution:

We know that, |x| can be defined as

|x| = $\left\{\begin{matrix} x;& x>0\\ -x;&x <0 \end{matrix}\right.$

Graph of y = |x| and x = 2 is

## Solving Absolute Value Examples

Let us see how to solve absolute value equations with the help of few examples.

### Absolute Value Equations Examples

Below are the examples on how to solve absolute value equations.

### Solved Examples

Question 1: Solve |x + 1| = 4
Solution:

The quantity inside the absolute value symbol can be equal to 4 or -4

x + 1 = 4 or x + 1 = - 4

Subtract 1 on both side of the given equation

x + 1 - 1 = 4 - 1 or x + 1 - 1 = -4 - 1

x = 3 or x = -5

So, the solution are x = 3 and x = -5

Question 2: Solve |2x - 3| = x - 5
Solution:

When solving this equation, you have to be careful when solving opposite of (x - 5)

2x - 3 = x - 5 or 2x - 3 = -(x - 5)

x - 3 = -5 or 2x -3 = -x + 5

x = -2 or 3x – 3 = 5

3x = 8

x = $\frac{8}{3}$

So, the solution is x = -2 and x = $\frac{8}{3}$

Question 3: Solve |x + 1| = 5.
Solution:

The quantity inside the absolute value symbol can be equal to 5 or -5
x + 1 = 5 or x + 1 =
-5
Subtract 1 on both
side of the given equation
x + 1 - 1 = 5 - 1 or
x + 1 - 1 = - 5 - 1
x = 4 or x = - 6
So, the solution are
x = 4 and x = - 6

### Practice Problem

Question: Solve |x + 1| = 5.