Absolute-Value Equations
An absolute-value equation like abs(2x - 5) = 9 asks for every x a fixed distance from zero after the bars are applied, so you split it into a positive case and a negative case and solve each.
Absolute value measures distance from zero, and distance has no direction, so abs(anything) = 9 means that 'anything' sits 9 units away from zero on either side. That gives two possibilities: the expression inside equals 9, or it equals -9. Solving an absolute-value equation means writing both of those equations and solving each one separately. Most of the time you get two real answers, because two different input values land the same distance from zero.
The method: isolate the absolute value expression on one side first, with nothing added, subtracted, multiplied, or divided outside the bars. Then write two equations, dropping the bars and setting the inside equal to the positive value, then equal to the negative value. Solve each normally. For abs(2x - 5) = 9, the two cases are 2x - 5 = 9 and 2x - 5 = -9, giving x = 7 and x = -2. Check both by plugging back in: 2(7) - 5 = 9, and 2(-2) - 5 = -9, which has absolute value 9. Both work.
This sits next to quadratic and radical equations in the same algebra cluster because all three can produce extra or missing solutions that need checking. Quadratics can give two roots from one equation the same way absolute value does, and radicals can introduce a solution that fails when you check it. The shared skill is isolating the tricky part first, then handling the two-branch or squaring step carefully, and always verifying answers in the original equation.
Key terms
- Absolute value
- The distance a number sits from zero on the number line, always zero or positive.
- Case split
- Rewriting abs(expression) = k as two separate equations, expression = k and expression = -k.
- Extraneous check
- Plugging a found solution back into the original equation to confirm it actually works.
- Isolate
- Rearranging an equation so the absolute-value expression stands alone on one side before splitting into cases.
Watch it explained
- How it worksHow does breaking into a positive case and a negative case capture both distances at once?
- Worked exampleWalk me through solving the absolute value of 2x - 5 equals 9 by splitting into two equations.
- Worked exampleFind every solution when 3 times the absolute value of x + 1, minus 4, equals 11.
- Common mistakeWhy isn't there any x making the absolute value of x - 3 come out to -2?
Check yourself
End of topic.