What Does Absolute Value Mean?
The absolute value of a number is its distance from zero on the number line, regardless of direction. We write it as |x|. For example, |5| = 5 and |−5| = 5 because both 5 and −5 are exactly five units away from zero.
|x| = x when x ≥ 0, and |x| = −x when x < 0.This second case often confuses students — it does not mean the result is negative. The negative sign in front of x simply flips the sign of a negative input, making the output positive. For instance, if x = −7, then |−7| = −(−7) = 7.
The Distance Interpretation
Geometrically, |a − b| represents the distance between points a and b on the number line. This distance interpretation is the key to understanding why absolute value equations and inequalities produce two cases. Because distance is always non-negative, |x| ≥ 0 for every real number x, and |x| = 0 only when x = 0.
Keeping this geometric meaning in mind will help you set up every absolute value problem correctly.
Solving Equations of the Form |ax + b| = c
The most fundamental absolute value equation has the form |ax + b| = c.
|expression| = c and c > 0, then the expression inside equals either c or −c.- Step 1 — Isolate
Isolate the absolute value on one side.
- Step 2 — Check the sign of c
Check that c is non-negative. If
c < 0, there is no solution because absolute value cannot be negative. Ifc = 0, solveax + b = 0for the single solution. - Step 3 — Split into two equations
If
c > 0, split into two equations:ax + b = candax + b = −c. - Step 4 — Solve each linear equation
Solve each linear equation separately.
|2x − 3| = 7Case 1: 2x − 3 = 7 → 2x = 10 → x = 5.
Case 2: 2x − 3 = −7 → 2x = −4 → x = −2.
Check: |2(5) − 3| = |7| = 7 and |2(−2) − 3| = |−7| = 7. Both solutions check out.
x = 5 or x = −2Always verify your answers by substituting back into the original equation — this habit catches errors and builds confidence.
Equations with Absolute Value on Both Sides
When an equation has absolute value expressions on both sides, such as |ax + b| = |cx + d|, the approach changes slightly.
ax + b = cx + d (the expressions are equal) and ax + b = −(cx + d) (the expressions are opposite).|3x − 1| = |x + 5|Case 1: 3x − 1 = x + 5 → 2x = 6 → x = 3.
Case 2: 3x − 1 = −(x + 5) → 3x − 1 = −x − 5 → 4x = −4 → x = −1.
Check x = 3: |8| = |8|. True.
Check x = −1: |−4| = |4|. True. Both solutions are valid.
x = 3 or x = −1A common mistake is writing four cases instead of two — you do not need to consider the signs of each absolute value independently, because the equation −(ax + b) = −(cx + d) reduces to the same thing as ax + b = cx + d. Two cases are always sufficient for this type of equation.
Solver AI automatically identifies the correct case structure for any absolute value equation and walks you through each case with clear steps.
Absolute Value Inequalities — Less Than
Absolute value inequalities of the form |ax + b| < c (or ≤) translate into a compound inequality.
The key insight comes from the distance interpretation: |expression| < c means the expression is within c units of zero.
|ax + b| < c is equivalent to −c < ax + b < c, provided c > 0. If c ≤ 0, the inequality |ax + b| < c has no solution (since absolute value is always non-negative), and |ax + b| ≤ 0 has a solution only when ax + b = 0.|4x − 2| ≤ 10Rewrite as a compound inequality: −10 ≤ 4x − 2 ≤ 10.
Add 2 to all three parts: −8 ≤ 4x ≤ 12.
Divide by 4: −2 ≤ x ≤ 3. The solution set is the closed interval [−2, 3].
[−2, 3]Think of it this way: "less than" keeps the variable between two bounds — it creates an "and" condition. A useful memory device: less thAND — the word "and" is hidden inside "less than."
On a number line, the solution is always a single connected interval centered around the point where the expression inside the absolute value equals zero.
Absolute Value Inequalities — Greater Than
Inequalities of the form |ax + b| > c (or ≥) work in the opposite direction from less-than inequalities. Here, |expression| > c means the expression is more than c units from zero, so it is either greater than c or less than −c.
ax + b > c OR ax + b < −c. This produces a disjoint union of two intervals. If c < 0, the inequality |ax + b| > c is satisfied by all real numbers (since absolute value is always ≥ 0 > c).|2x + 1| > 5Case 1: 2x + 1 > 5 → 2x > 4 → x > 2.
Case 2: 2x + 1 < −5 → 2x < −6 → x < −3.
(−∞, −3) ∪ (2, ∞)On the number line, you shade two rays going in opposite directions. The memory device here: greatOR — the word "or" is hidden inside "greater than." The solution always has two separate pieces.
This "or" structure is what distinguishes greater-than absolute value inequalities from less-than ones, and keeping this distinction clear is essential for getting the correct answer on exams and homework.
Common Mistakes and Extraneous Solutions
Absolute value problems are fertile ground for errors, so here are the most common mistakes to watch out for.
If you have 3|x − 2| + 4 = 13, you must first subtract 4 and divide by 3 to get |x − 2| = 3 before splitting cases.
The equation |x + 1| = −4 has no solution. Never split into cases if the right side is negative — immediately write "no solution."
Remember: less thAND, greatOR.
Extraneous solutions arise especially when you square both sides or when absolute value expressions are nested. Always substitute your answers back into the original equation.
When writing the negative case of |3x − 7| = |x + 2|, the second case is 3x − 7 = −x − 2, not 3x − 7 = −x + 2. The negative sign applies to the entire expression.
Building the habit of checking every solution in the original equation is the single most effective way to avoid losing points. Solver AI flags extraneous solutions automatically and shows exactly why a candidate solution fails, helping you develop this critical-checking mindset.