Absolute value equations can be solved by squaring both sides of the equation. In this way, the expression inside the absolute value sign will always be positive. Then we can expand the expressions and simplify them. Finally, the obtained equation can be solved with any applicable method.
Here, we will look at 10 examples of equations with absolute value. In addition, you will be able to test your skills with some practice problems.
How 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.
10 Examples of equations with absolute value
Each of the following examples has a detailed solution. However, try to solve the problems yourself before looking at the solution.
EXAMPLE 1
Solve the equation $latex |x-2|=4$.
Solution
To make sure that the left-hand side of the equation is positive, we can square both sides of the equation:
$latex |x-2|^2=4^2$
Now, we can replace the absolute value signs with parentheses:
$latex (x-2)^2=4^2$
Expanding the parentheses and simplifying, we have:
$latex (x-2)^2=4^2$
$latex x^2-4x+4=16$
$latex x^2-4x-12=0$
We can solve the quadratic equation by factoring:
$latex x^2-4x-12=0$
$latex (x-6)(x+2)=0$
The solutions are $latex x=6$ and $latex x=-2$.
EXAMPLE 2
Find the solution to the equation $latex |x+4|=5$.
Solution
Let’s square both sides to make sure the left-hand side is positive:
$latex |x+4|^2=5^2$
Using parentheses instead of absolute value signs, we have:
$latex (x+4)^2=5^2$
Now, we simplify by expanding the parentheses:
$latex (x+4)^2=5^2$
$latex x^2+8x+16=25$
$latex x^2+8x-9=0$
Solving by factorization, we have:
$latex x^2+8x-9=0$
$latex (x+9)(x-1)=0$
The solutions are $latex x=-9$ and $latex x=1$.
EXAMPLE 3
Solve the equation $latex |3-x|=6$.
Solution
Squaring both sides, we have:
$latex |3-x|^2=6^2$
$latex (3-x)^2=6^2$
Simplifying the equation, we have:
$latex (3-x)^2=6^2$
$latex 9-6x+x^2=36$
$latex x^2-6x-27=0$
Now, we solve the equation by factoring:
$latex x^2-6x-27=0$
$latex (x-9)(x+3)=0$
The solutions are $latex x=9$ and $latex x=-3$.
EXAMPLE 4
Find the solution to the equation $latex |2x+1|=5$.
Solution
By squaring both sides of the equation, we have:
$latex |2x+1|^2=5^2$
$latex (2x+1)^2=5^2$
Expanding the parentheses and simplifying the equation, we have:
$latex (2x+1)^2=5^2$
$latex 4x^2+4x+1=25$
$latex 4x^2+4x-24=0$
Solving by factorization, we have:
$latex 4(x^2+x-6)=0$
$latex 4(x+3)(x-2)=0$
The solutions are $latex x=-3$ and $latex x=2$.
EXAMPLE 5
What is the solution to the equation $latex |3x+2|=8$?
Solution
When we square both sides of the equation, we have:
$latex |3x+2|^2=8^2$
$latex (3x+2)^2=8^2$
Expanding the parentheses and simplifying, we have:
$latex (3x+2)^2=8^2$
$latex 9x^2+12x+4=64$
$latex 9x^2+12x-60=0$
Factoring and solving, we have:
$latex 9x^2+12x-60=0$
$latex (3x+10)(3x-6)=0$
The solutions are $latex x=-\frac{10}{3}$ and $latex x=2$.
EXAMPLE 6
Find the solution to the equation $latex |5x-3|=7$.
Solution
When we square both sides, we have:
$latex |5x-3|^2=7^2$
$latex (5x-3)^2=7^2$
Expanding the parentheses on the left-hand side and simplifying, we have:
$latex (5x-3)^2=7^2$
$latex 25x^2-30x+9=49$
$latex 25x^2-30x-40=0$
Factoring and solving, we have:
$latex 25x^2-30x-40=0$
$latex (5x-10)(5x+4)=0$
The solutions are $latex x=2$ and $latex x=-\frac{4}{5}$.
EXAMPLE 7
Find the solution to the equation $latex |x+1|=|x-3|$.
Solution
Squaring both sides of the equation, we have:
$latex |x+1|^2=|x-3|^2$
$latex (x+1)^2=(x-3)^2$
Expanding both parentheses and simplifying by combining like terms, we have:
$latex (x+1)^2=(x-3)^2$
$latex x^2+2x+1=x^2-6x+9$
$latex 8x=8$
In this case, we have a linear equation that can be easily solved:
$latex 8x=8$
$latex x=1$
The only solution is $latex x=1$.
EXAMPLE 8
Find the solution to the equation $latex |x-4|=|6-x|$.
Solution
We start by squaring both sides of the equation:
$latex |x-4|^2=|6-x|^2$
$latex (x-4)^2=(6-x)^2$
We expand both parentheses and simplify the equation:
$latex (x-4)^2=(6-x)^2$
$latex x^2-8x+16=36-12x+x^2$
$latex 4x=20$
We can easily solve the linear equation:
$latex 4x=20$
$latex x=5$
The only solution is $latex x=5$.
EXAMPLE 9
Solve the equation $latex |2x-1|=|x|$.
Solution
Squaring both sides, we have:
$latex |2x-1|^2=|x|^2$
$latex (2x-1)^2=(x)^2$
We expand the parentheses and simplify as follows:
$latex (2x-1)^2=x^2$
$latex 4x^2-4x+1=x^2$
$latex 3x^2-4x+1=0$
Solving by factorization, we have:
$latex 3x^2-4x+1=0$
$latex (3x-1)(x-1)=0$
The solutions are $latex x=\frac{1}{3}$ and $latex x=1$.
EXAMPLE 10
Solve the equation $latex |2x-1|=|4x+3|$.
Solution
Squaring both sides of the equation, we have:
$latex |2x-1|^2=|4x+3|^2$
$latex (2x-1)^2=(4x+3)^2$
Expanding the parentheses and simplifying, we have:
$latex (2x-1)^2=(4x+3)^2$
$latex 4x^2-4x+1=16x^2+24x+9$
$latex 12x^2+28x+8=0$
Solving by factorization, we have:
$latex 4(3x^2+7x+2)=0$
$latex (3x+1)(x+2)=0$
The solutions are $latex x=-\frac{1}{3}$ and $latex x=-2$.
5 Practice problems of equations with absolute value
Test your knowledge of absolute value equations to solve the following practice problems. You can use the examples with answers shown above as a guide.
See also
Interested in learning more about absolute value equations and inequalities? You can take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
