# How to Solve Equations with Absolute Value? – Step-by-step

To solve equations that use the absolute value function, we can start by squaring both sides of the equation. This will ensure that we get a positive value in the expression that has the absolute value. We can then solve the equation normally by removing the absolute value symbols.

In this article, we will learn how to solve absolute value equations step-by-step. In addition, we will look at some examples in which we will apply this process.

##### ALGEBRA

Relevant for

Learning to solve equations with absolute value.

See steps

##### ALGEBRA

Relevant for

Learning to solve equations with absolute value.

See steps

## Steps to solve absolute value equations

The absolute value function $latex f(x)=|x|$, also called the modulus of x, can be defined as the magnitude of x. For example:

$latex |-2|=2~~$ and $latex ~~|2|=2$

To solve equations with absolute value, we can follow the following steps:

Step 1: Square both sides of the equation.

This will make sure that the expression that has the absolute value is positive since the absolute value function represents the magnitude.

Step 2: Change the absolute value signs to parentheses.

Step 3: Expand and simplify the parentheses and the squared expressions.

Step 4: Solve the quadratic equation obtained.

Note: If you need to review how to solve quadratic equations, you can visit our article: Solving Quadratic Equations – Methods and Examples.

## Equations with absolute value – Examples with answers

The following examples are solved by applying the process of solving absolute value equations seen above. Try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

Solve the equation $latex |x-1|=4$.

Step 1: Squaring both sides of the equation, we have:

$latex |x-1|^2=4^2$

Step 2: Now, we change the absolute value signs to parentheses:

$latex (x-1)^2=4^2$

Step 3: Simplifying, we have:

$latex (x-1)^2=4^2$

$latex x^2-2x+1=16$

$latex x^2-2x-15=0$

Step 4: We can solve the equation by factoring:

$latex x^2-2x-15=0$

$latex (x-5)(x+3)=0$

The solutions are $latex x=5$ and $latex x=-3$.

### EXAMPLE 2

Find the solutions to the equation $latex |4-x|=2$.

Step 1: We start by squaring the entire equation:

$latex |4-x|^2=2^2$

Step 2: Changing the absolute value signs to parentheses, we have:

$latex (4-x)^2=2^2$

Step 3: Expanding the parentheses and simplifying, we have:

$latex (4-x)^2=2^2$

$latex 16-8x+x^2=4$

$latex x^2-8x+12=0$

Step 4: Solving by factoring, we have:

$latex x^2-8x+12=0$

$latex (x-6)(x-2)=0$

The solutions are $latex x=6$ and $latex x=2$.

### EXAMPLE 3

Solve the equation $latex |3x+1|=4$.

Step 1: When we square both sides of the equation, we have:

$latex |3x+1|^2=4^2$

Step 2: We use parentheses instead of absolute value signs:

$latex (3x+1)^2=4^2$

Step 3: Expanding the parentheses and simplifying, we have:

$latex (3x+1)^2=4^2$

$latex 9x^2+6x+1=16$

$latex 9x^2+6x-15=0$

Step 4: We solve by factorization:

$latex 9x^2+6x-15=0$

$latex (3x+5)(3x-3)=0$

The solutions are $latex x=-\frac{5}{3}$ and $latex x=1$.

### EXAMPLE 4

What is the solution to the equation $latex |5x-3|=7$?

Step 1: When we square both sides, we have:

$latex |5x-3|^2=7^2$

Step 2: We use parentheses instead of absolute value signs:

$latex (5x-3)^2=7^2$

Step 3: Expanding the parentheses and simplifying, we have:

$latex (5x-3)^2=7^2$

$latex 25x^2-30x+9=49$

$latex 25x^2-30x-40=0$

Step 4: We can solve the equation by factoring:

$latex 25x^2-30x-40=0$

$latex (5x-10)(5x+4)=0$

The solutions are $latex x=2$ and $latex x=-\frac{4}{5}$.

### EXAMPLE 5

Solve the equation $latex |x+1|=|x-3|$.

In this case, we have an equation with absolute value on both sides of the equal sign, but we can use the same process to solve.

Step 1: By squaring both sides of the equation, we have:

$latex |x+1|^2=|x-3|^2$

Step 2: Substituting the absolute value signs for parentheses, we have:

$latex (x+1)^2=(x-3)^2$

Step 3: Expanding the parentheses and simplifying, we have:

$latex (x+1)^2=(x-3)^2$

$latex x^2+2x+1=x^2-6x+9$

$latex 8x=8$

Step 4: In this case, we have a linear equation, which we can easily solve:

$latex 8x=8$

$latex x=1$

The equation has only one solution, $latex x=1$.

### EXAMPLE 6

Solve the equation $latex |x-4|=|6-x|$.

Step 1: When we square both sides of the equation, we have:

$latex |x-4|^2=|6-x|^2$

Step 2: We use parentheses instead of absolute value signs:

$latex (x-4)^2=(6-x)^2$

Step 3: Expanding the parentheses and simplifying, we have:

$latex (x-4)^2=(6-x)^2$

$latex x^2-8x+16=36-12x+x^2$

$latex 4x=20$

Step 4: We can easily solve the linear equation:

$latex 4x=20$

$latex x=5$

The only solution is $latex x=5$.

## Equations with absolute value – Practice problems

Find the solution to the following practice problems to apply everything learned about quadratic inequalities.

#### Find the solution to the equation $latex |2x-2|=|2-3x|$.  