Absolute value equations look confusing at first, but once you understand the basic idea, they become very simple.
In this blog, you will learn what absolute value means, how to solve absolute value equations, step-by-step methods, and examples for practice.
What Is Absolute Value? (Easy Explanation)
The absolute value of a number means its distance from 0 on the number line.
Distance is always positive, so absolute value is also always positive.
Symbol:
Absolute value is written using vertical bars:
|x|
Examples:
- |5| = 5
- |-5| = 5
- |0| = 0
So the absolute value removes the negative sign.
Types of Absolute Value Equations
There are three main types:
- |expression| = positive number → Two solutions
- |expression| = 0 → One solution
- |expression| = negative number → No solution
Let’s learn these one by one.
1. Solving Equations Like: |x| = a (a > 0)
If the absolute value equals a positive number, then:
|x| = a
means
x = a or x = –a
Example 1:
Solve:
|x| = 7
Step 1: Remove the bars and create two equations:
x = 7
x = –7
Final Answer:
x = 7 or x = –7
2. Solving Equations Like: |x| = 0
Only one value has absolute value 0: x = 0
Example 2:
Solve:
|x| = 0
Answer:
x = 0
3. Solving Equations Like: |x| = –a (Negative Number)
Absolute values are never negative.
So if the equation looks like this:
|x| = –3
There is no solution.
Example 3:
Solve:
|x| = –9
Answer:
No solution.
How to Solve Absolute Value Equations With Expressions
Now let’s look at equations that have algebra inside the absolute value.
Case 1: |x + b| = c
If c > 0, create two equations:
x + b = c
x + b = –c
Example 4:
Solve:
|x + 3| = 5
Step 1: Create two equations:
- x + 3 = 5
- x + 3 = –5
Step 2: Solve both:
- x = 5 – 3 = 2
- x = –5 – 3 = –8
Final Answer:
x = 2 or x = –8
Case 2: a|x + b| = c
First divide both sides by a.
Example 5:
Solve:
2|x – 4| = 10
Step 1: Divide both sides by 2
|x – 4| = 5
Step 2: Create two equations:
- x – 4 = 5
- x – 4 = –5
Step 3: Solve:
- x = 9
- x = –1
Final Answer:
x = 9 or x = –1
Harder Example (But Easy Explanation)
Example 6:
Solve:
|2x – 6| = 14
Step 1: Create two equations:
- 2x – 6 = 14
- 2x – 6 = –14
Step 2: Solve both:
- 2x = 20 → x = 10
- 2x = –8 → x = –4
Final Answer:
x = 10 or x = –4
When Absolute Value Has Variables on Both Sides
Example 7:
Solve:
|x – 5| = |2x – 3|
Here we make two cases:
Case 1:
x – 5 = 2x – 3
Solve:
–5 + 3 = x
x = –2
Case 2:
x – 5 = –(2x – 3)
x – 5 = –2x + 3
3x = 8
x = 8/3
Final Answer:
x = –2 or x = 8/3
Quick Shortcut Table
| Equation Type | Number of Solutions | Method |
|---|---|---|
| |x| = positive number | 2 solutions | x = a, x = –a |
| |x| = 0 | 1 solution | x = 0 |
| |x| = negative number | No solution | stop |
| a|x + b| = c | 2 or 1 or 0 | divide → create two equations |
| |expr1| = |expr2| | 2 cases | expr1 = expr2 and expr1 = –expr2 |
Practice Questions
Try these yourself:
- |x| = 12
- |x – 9| = 3
- 4|x + 1| = 16
- |2x + 5| = –7
- |3x – 6| = |x + 4|
Want solutions for the practice questions?
Just say “give the answers” and I will send them.
If you found this guide on How to Solve Absolute Value Equations Easily helpful, please share it with your friends, classmates, or anyone who might benefit from it!
Article End Here
Read More Like This Click Here
