How to Solve Absolute Value Equations: Step-by-Step Guide + Calculator

Introduction: What Are Absolute Value Equations?

An absolute value equation is any equation that has a variable inside absolute value bars, such as |x| = 5 or |2x - 3| = 7. Learning how to solve absolute value equations is essential in algebra because they appear in:

distance problems on the number line
piecewise and “V-shaped” graphs
inequalities and optimization questions

In this guide, you will learn step by step how to solve absolute value equations, check your answers, and even interpret solutions using a calculator or graph.

Key Properties of Absolute Value You Must Know

The absolute value of a real number x, written |x|, is its distance from 0 on the number line.

Formally:

If x ≥ 0, then |x| = x.
If x < 0, then |x| = -x (the opposite of x).

Important consequences for equations:

Absolute values are always nonnegative: |x| ≥ 0 for all real x.
If |x| = a and a < 0, then there is no solution.
If |x| = a and a ≥ 0, then x = a or x = -a.

Solving Basic Absolute Value Equations: |x| = a

The most basic type of absolute value equation is |x| = a, where a is a constant.

General rule

If a > 0, then:

|x| = a has two solutions:
x = a or x = -a.

If a = 0, then |x| = 0 has one solution: x = 0.

If a < 0, then |x| = a has no solution.

Example 1: Solve |x| = 6

Recognize that 6 > 0.
Write two equations: x = 6 or x = -6.
Solutions: x = 6, x = -6.

Example 2: Solve |x| = -4

Absolute value cannot be negative.
So |x| = -4 has no real solution.

Shifts and Translations: Solving |x - c| = a

Now the expression inside the absolute value is not just x, but x - c. The idea is the same: the quantity inside can be positive or negative, so we split into two cases.

General strategy

To solve |x - c| = a with a ≥ 0:

Write two equations:
- x - c = a
- x - c = -a
Solve each equation for x.

Example 3: Solve |x - 3| = 5

Set up cases:
- x - 3 = 5
- x - 3 = -5
Solve each:
- x - 3 = 5 ⇒ x = 8
- x - 3 = -5 ⇒ x = -2
Solutions: x = 8 and x = -2.

Example 4: Solve |x + 4| = 2

Remember x + 4 is the same as x - (-4), so c = -4.

x + 4 = 2 or x + 4 = -2
Solve:
- x = -2
- x = -6

Equations with Coefficients: Solving |ax + b| = c

More generally, you will often see equations like |ax + b| = c, where a, b, and c are constants.

Step-by-step method

Isolate the absolute value expression on one side (if necessary).
Ensure the other side is just a constant c.
If c < 0, conclude there is no solution.
If c ≥ 0, write two linear equations:
- ax + b = c
- ax + b = -c
Solve each for x.

Example 5: Solve |2x - 1| = 7

Write two equations:
- 2x - 1 = 7
- 2x - 1 = -7
Solve:
- 2x - 1 = 7 ⇒ 2x = 8 ⇒ x = 4
- 2x - 1 = -7 ⇒ 2x = -6 ⇒ x = -3
Solutions: x = 4, x = -3.

Example 6: Solve 3|x + 2| = 9

Isolate the absolute value:
3|x + 2| = 9 ⇒ |x + 2| = 3
Set up cases:
- x + 2 = 3
- x + 2 = -3
Solve:
- x = 1
- x = -5

Multiple Absolute Values: Solving |ax + b| = |cx + d|

When both sides have absolute values, you can still use case splitting.

General idea

Use the fact that if |A| = |B|, then either:

A = B
A = -B

Here A = ax + b and B = cx + d.

Example 7: Solve |2x - 3| = |x + 1|

Set up the two cases:
- Case 1: 2x - 3 = x + 1
- Case 2: 2x - 3 = -(x + 1)
Solve Case 1:
2x - 3 = x + 1 ⇒ 2x - x = 4 ⇒ x = 4
Solve Case 2:
2x - 3 = -x - 1 ⇒ 2x + x = 2 ⇒ 3x = 2 ⇒ x = 2/3
Check both solutions in the original equation (we will discuss checking in detail later).
Final solutions: x = 4 and x = 2/3.

Special Cases: No Solution and Infinitely Many Solutions

No solution

A typical “no solution” situation appears when the isolated absolute value equals a negative number.

Example: |3x - 1| = -2.

The left side is always ≥ 0.
The right side is -2, which is < 0.
They can never be equal → no solution.

Infinitely many solutions

This can happen if both sides are the same absolute value expression.

Example: |2x + 5| = |2x + 5|.

This is true for every real number x, so the solution set is “all real numbers”.

Checking Solutions: Why Plugging Back Is Essential

When you solve absolute value equations using case splitting, you sometimes create extraneous solutions (values that satisfy the case equation but not the original equation).

To avoid mistakes, always substitute your answers back into the original equation.

Example 8: Check a solution

Suppose we solved |2x - 1| = 7 and found x = 4.

Plug in: |2(4) - 1| = |8 - 1| = |7| = 7 ✓
So x = 4 is a valid solution.

Using a Calculator or Online Solver for Absolute Value Equations

You can use graphing calculators or online tools to help solve absolute value equations—but you should still understand the algebraic steps.

How to enter absolute values on a calculator

Many calculators have an abs( ) function (often under MATH → NUM).
To solve |2x - 1| = 7, you can graph:
- y1 = abs(2x - 1)
- y2 = 7
Then use “intersect” to find the x-values where y1 = y2.

Using an online absolute value equation solver

Enter the equation exactly as written, using | or abs().
Let the solver simplify and produce the solution set.
Compare with your own algebraic work to check if you solved the absolute value equation correctly.

Graphical Method: Visualizing Solutions to Absolute Value Equations

Graphs give you an intuitive picture of how to solve absolute value equations. The basic graph of y = |x| is a V-shape with its vertex at the origin.

Single absolute value on one side

For an equation like |2x - 1| = 7:

Graph y1 = |2x - 1| (a V-shaped graph).
Graph y2 = 7 (a horizontal line).
The x-coordinates of their intersection points are the solutions.

Absolute value on both sides

For |2x - 3| = |x + 1|, graph:

y1 = |2x - 3|
y2 = |x + 1|

Again, the intersection x-values are the solutions. You should get x = 4 and 2/3, matching the algebraic method.

Common Pitfalls and Quick Tips for Absolute Value Equations

Always isolate the absolute value first before splitting into cases.
Check whether the number on the other side is negative; if so, there is no solution.
Remember that |x - c| = a leads to x = c + a and x = c - a.
When both sides have absolute values, use A = B and A = -B.
Always check your answers by plugging them into the original equation.

Practice Problems: Beginner to Intermediate

Try each problem on your own first, then compare with the worked solutions.

Beginner

Problem 1: Solve |x| = 9.
Solution:
x = 9 or x = -9.
Problem 2: Solve |x - 4| = 2.
Solution:
Cases: x - 4 = 2 or x - 4 = -2.
So x = 6 or x = 2.
Problem 3: Solve |x + 3| = 0.
Solution:
Only when the inside is 0: x + 3 = 0 ⇒ x = -3.

Intermediate

Problem 4: Solve |3x - 2| = 4.
Solution:
Cases: 3x - 2 = 4 or 3x - 2 = -4.
From 3x - 2 = 4: 3x = 6 ⇒ x = 2.
From 3x - 2 = -4: 3x = -2 ⇒ x = -2/3.
Problem 5: Solve 2|x + 1| - 5 = 3.
Solution:
Isolate: 2|x + 1| - 5 = 3 ⇒ 2|x + 1| = 8 ⇒ |x + 1| = 4.
Cases: x + 1 = 4 or x + 1 = -4.
Solutions: x = 3, x = -5.
Problem 6: Solve |2x + 1| = |x - 5|.
Solution:
Let A = 2x + 1, B = x - 5.
Case 1: A = B ⇒ 2x + 1 = x - 5 ⇒ x = -6.
Case 2: A = -B ⇒ 2x + 1 = -(x - 5) = -x + 5.
Then 2x + x = 5 - 1 ⇒ 3x = 4 ⇒ x = 4/3.
Solutions: x = -6, x = 4/3.

Next Step

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