The absolute value of a number is its distance from zero, so it is never negative.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two cases
|x| = 5 means x is 5 units from zero, which happens at 5 and at −5.
Isolate first
Get the absolute value alone before splitting into cases, or the split is wrong.
When there is no answer
|x| = −3 has no solution. A distance cannot be negative.
An absolute value equation splits in two
|x − 3| = 5 means the distance from x to 3 is 5, which happens on either side: x − 3 = 5 or x − 3 = −5. So x = 8 or x = −2. Every absolute value equation becomes two ordinary ones.
Isolate the absolute value first
For 2|x − 3| + 1 = 11, get to |x − 3| = 5 before splitting. Splitting while other terms are still attached produces two wrong equations, and the error is hard to spot afterwards.
When there is no solution
|x − 3| = −5 has no solution, because a distance can never be negative. Checking for a negative right-hand side before splitting saves the work of solving two impossible equations.
Check both answers
Substitute each solution back into the original. Extraneous solutions can appear if the isolation step was mishandled, and only substitution into the original equation will reveal them.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Dev Split Into Cases
Dev solves |x − 2| = 6.
- Step 1
The absolute value is already alone.
Step 4: Your Turn
Practice makes it stick.
The Positive Case
Problem 1 of 2
|x| = 9. What is the positive solution?
The Impossible One
Problem 2 of 2
|x| = −4. How many solutions?
Both Directions
1 of 8
|x| = 7. Positive solution?
2 of 8
|x| = 7. Negative solution?
3 of 8
|x − 3| = 5. What is the larger solution?
4 of 8
|x − 3| = 5. What is the smaller solution?
5 of 8
|x| = 0. How many solutions?
6 of 8
|x + 1| = 4. What is the larger solution?
7 of 8
|x| = −10. How many solutions?
8 of 8
Select every equation that has exactly two solutions.
Step 5: Quick Check
Show what you know.
Question 1 of 1
|x − 4| = 3. What is the larger solution?
What You Learned
- Absolute value is distance from zero and is never negative.
- An absolute value equation usually splits into two cases.
- Equal to zero gives one solution; equal to a negative gives none.