Calculators/Mathematics/Absolute value — equations and inequalities

Absolute value — equations and inequalities

Q: When does an absolute-value inequality have no solutions?

|ax+b| < c has no solutions when c ≤ 0 (except c = 0 for "≤", which gives a single point) — because an absolute value is never negative.

|ax + b| ⋛ c · solutions and intervals

Absolute value — equations and inequalities — illustration

|x +|

Solution

x = -1 or x = 5

|ax+b| = c means ax+b = c or ax+b = −c. For "less than" you get the interval between the roots, for "greater than" — two intervals outside them. An absolute value is never negative.

The absolute value $|ax+b|$ is the distance of an expression from zero — always non-negative. How you solve it depends on the type of relation:

Equation $|ax+b|=c$ : for $c>0$ it gives two solutions ( $ax+b=c$ or $ax+b=-c$ ), for $c=0$ one, for $c<0$ none.
Inequality $|ax+b|<c$ : the solution is the interval between the roots ( $-c<ax+b<c$ ).
Inequality $|ax+b|>c$ : the solution is the union of two intervals on the outside.

The calculator handles every relation ( $=,<,\le,>,\ge$ ) and the edge cases (e.g. $c<0$ ).

Examples

|x − 2| = 3

→ x = −1 or x = 5

|x − 2| < 3

→ x ∈ (−1, 5)

|x − 2| > 3

→ x ∈ (−∞, −1) ∪ (5, +∞)

Frequently asked questions

Why does |ax+b| = c give two solutions?+

Because the inner expression can equal c or −c — both have the same absolute value. Hence two equations and (for c > 0) two roots.

When does an absolute-value inequality have no solutions?+

|ax+b| < c has no solutions when c ≤ 0 (except c = 0 for "≤", which gives a single point) — because an absolute value is never negative.

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