The absolutely value |x| can be defined as

|x| = $\left\{\begin{matrix}

x; & x>0\\

-x;&x <0

\end{matrix}\right.$

Related Calculators | |

Absolute Value Equation Solver | Absolute Value Calculator |

absolute mean deviation calculator | Absolute Pressure Calculator |

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

### Solving Absolute Value Equations and Inequalities

### 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: **
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

Let us see how to solve absolute value equations with the help of few examples.### Absolute Value Equations Examples

### Solved Examples

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

** Solution: **

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

** Solution: **

**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.

Given below are some of the practice problems om absolute value equations. ### Practice Problems

**Question 1: **Solve |x + 1| = 6

**Question 2: **Solve |x + 1| = 7

$\Rightarrow$ 3x = 15

$\Rightarrow$ x = 5

$\Rightarrow$ 3x = - 5

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

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

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].

$\Rightarrow$ -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].

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

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

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

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

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 = $\frac{8}{3}$

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

Related Topics | |

Math Help Online | Online Math Tutor |