Solving |x − p| = q
Simple Explanation
|x − p| = q asks "which values of x are exactly q units away from p?" If q > 0, there are always two such values: x = p + q and x = p − q. If q = 0, there is exactly one (x = p). If q < 0, there is no solution, since distance can never be negative.
Why Do We Need It?
This "two symmetric solutions" pattern shows up constantly — whenever a problem asks for values within an exact distance of a target, it is really an absolute value equation in disguise.
See It
A V-shaped graph with vertex at (3, -5), crossing the x-axis at x = -2 and x = 8
Formula
Solving |x − p| = q
If q ≥ 0: x − p = q or x − p = −q, i.e. x = p + q or x = p − q. If q < 0: no solution.
Since |x − p| measures a distance (from x to p) and distance is never negative, this equation asks "which x-values are exactly q units away from p?" — there are exactly two, one on each side of p, unless q is negative.
- x
- — the unknown being solved for
- p
- — the reference point the distance is measured from
- q
- — the required distance from p (must be ≥ 0 for a solution to exist)
When to use it: Whenever you need to solve an equation with a single absolute value expression set equal to a number.
Worked Example
Solve an absolute value equation
Solve |x − 3| = 5.
Why Does This Work?
|x − p| represents the distance between x and p on the number line, and distance never depends on direction — so there are exactly two points at any given positive distance from p, one on each side, which is exactly what splitting into the + and − cases captures.
Real-Life Example
Acceptable manufacturing tolerance
A bolt must have a diameter within 0.5 mm of the target 10 mm, described by |diameter − 10| = 0.5 at the boundary of acceptability.
Solving gives diameter = 10.5 mm or 9.5 mm — the two exact boundary values, found the same way as any absolute value equation.
Practice
Solve |x + 2| = 7.
MediumHow many solutions does |x − 4| = −3 have?
EasyCommon mistake
Forgetting the negative case entirely and reporting only one solution — an absolute value equation with q > 0 always has two solutions, not one.
Quick Review
- |x − p| = q splits into x − p = q or x − p = −q, when q ≥ 0.
- q > 0 gives two solutions; q = 0 gives one; q < 0 gives no solution.
- Geometrically: the two solutions are the points exactly q units on either side of p.