Absolute Value Equation Calculator
Solve equations with absolute values - get both solutions explained
Format: |expression| = number
Understanding Absolute Value Equations
The absolute value of a number is its distance from zero on the number line, always non-negative. When solving absolute value equations, we must consider both the positive and negative cases.
Key Concepts
Definition: |x| = x if x ≥ 0, and |x| = -x if x < 0
Two Cases: If |A| = B (where B > 0), then A = B or A = -B
Special Cases:
- If |A| = 0, then A = 0 (one solution)
- If |A| = negative number, no solution exists
Example
Solve: |2x - 1| = 5
Step 1: Set up two cases
Case 1: 2x - 1 = 5
Case 2: 2x - 1 = -5
Step 2: Solve Case 1
2x = 6, so x = 3
Step 3: Solve Case 2
2x = -4, so x = -2
Solutions: x = 3 or x = -2
Why Two Solutions?
Since absolute value measures distance from zero, two different numbers can have the same absolute value. For example, both 3 and -3 have an absolute value of 3, because both are 3 units from zero.
Frequently Asked Questions
What is absolute value?
Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative number. For example, |5| = 5 and |-5| = 5.
Why do absolute value equations have two solutions?
Because both a positive and negative number can have the same absolute value. If |x| = 5, then x could be 5 or -5, since both are 5 units from zero.
When does an absolute value equation have no solution?
When the absolute value equals a negative number. Since absolute value is always non-negative, equations like |x| = -3 have no solution.
When does an absolute value equation have one solution?
When the absolute value equals zero, like |x - 3| = 0. This means x - 3 = 0, so x = 3 is the only solution.
How do I check my solutions?
Substitute each solution back into the original equation. Both solutions should make the equation true. For |2x - 1| = 5, check x = 3: |2(3) - 1| = |5| = 5 ✓
Can absolute value equations have more than two solutions?
Simple absolute value equations of the form |ax + b| = c have at most two solutions. More complex equations with multiple absolute values can have more solutions.
What's the difference between |x + 2| and -(x + 2)?
|x + 2| is always non-negative (the absolute value), while -(x + 2) can be positive or negative (it's the opposite of x + 2). They're only equal when x + 2 ≤ 0.
Do I always need to consider two cases?
Yes, unless the right side is zero or negative. If it's zero, solve one equation. If it's negative, there's no solution. For positive values, always consider both cases.