Absolute Value Equations & Inequalities

Q: When does an absolute value equation have no solution?

When it equals a negative number, like $|x| = -3$. Absolute value is a distance and can never be negative.

Absolute value measures distance from zero, so an equation like $|2x - 1| = 7$ asks 'what makes the inside 7 units from zero?' The answer is two cases: the inside equals $7$ or $-7$ .

Inequalities split the same way but become compound statements. 'Less than' gives an 'and' (a band between two values); 'greater than' gives an 'or' (two outer pieces).

Solving the equation with two cases

For $|A| = b$ with $b \geq 0$ , write $A = b$ or $A = -b$ and solve each. The two solutions are the two inputs that land $b$ units from zero.

If $b$ is negative there is no solution — absolute value can never be negative. Always check the right side before splitting.

Inequalities: and versus or

'Less than' ( $|A| < b$ ) means the inside is within $b$ of zero: $-b < A < b$ , a single band — an 'and.'

'Greater than' ( $|A| > b$ ) means the inside is farther than $b$ from zero: $A < -b$ or $A > b$ , two pieces — an 'or.' The memory aid is 'less-thand' and 'great-or.'

Worked examples

Example 1: an absolute value equation

Solve $|2x - 1| = 7$ .

Split into two cases

2x - 1 = 7 ;\text{or}; 2x - 1 = -7

Solve each

2x = 8 ;\text{or}; 2x = -6

Divide by 2

x = 4 ;\text{or}; x = -3

Answer: $x = 4$ or $x = -3$

Example 2: a 'less than' inequality

Solve $|x| < 5$ .

Less than becomes a band

-5 < x < 5

Answer: $-5 < x < 5$

Example 3: a 'greater than' inequality

Solve $|x - 3| > 4$ .

Greater than becomes two outer pieces

x - 3 < -4 ;\text{or}; x - 3 > 4

Add 3 to each side

x < -1 ;\text{or}; x > 7

Answer: $x < -1$ or $x > 7$

Try one yourself

Common questions

When does an absolute value equation have no solution?

When it equals a negative number, like $|x| = -3$ . Absolute value is a distance and can never be negative.

How do I remember 'and' versus 'or'?

'Less-thand' — less than gives an AND (a band between two values). 'Great-or' — greater than gives an OR (two outer pieces).

Do I always split into two cases?

For equations and inequalities, yes — one case for the positive value inside, one for the negative. Just isolate the absolute value first.

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Absolute Value Equations & Inequalities

Solving the equation with two cases

Inequalities: and versus or

Worked examples

Example 1: an absolute value equation

Example 2: a 'less than' inequality

Example 3: a 'greater than' inequality

Try one yourself

Common questions

Keep learning

Two-Step Inequalities

Piecewise, Step & Absolute Value Functions

Equations of Linear Functions