Solving a linear inequality
A linear inequality uses an inequality symbol (<, >, ≤, or ≥) instead of an equals sign, so its solution set is usually a whole range of numbers rather than a single value. You solve it with the very same balance steps as an equation, with one crucial exception: multiplying or dividing both sides by a negative number reverses the inequality symbol. Adding, subtracting, and multiplying or dividing by a positive all leave the symbol alone.
Solve -2x + 1 < 9 Subtract 1 from both sides (symbol unchanged): -2x < 8 Divide both sides by -2 -> FLIP the symbol: x > -4 Solution set: all x greater than -4. Interval notation: (-4, infinity). Quick test x = 0: -2(0) + 1 = 1 < 9 TRUE, and 0 > -4. Consistent.
Compound inequalities: and / or
A compound inequality joins two conditions. An and statement like -1 ≤ 2x + 3 < 7 must hold on both counts at once; solve it by doing the same step to all three parts. An or statement, like x < -2 or x > 5, is satisfied when either piece is true, giving a solution in two separate chunks.
Solve -1 <= 2x + 3 < 7 (an 'and') Subtract 3 from all three parts: -4 <= 2x < 4 Divide all three parts by 2 (positive, no flip): -2 <= x < 2 Solution set: all x with -2 <= x < 2. Interval notation: [-2, 2).
Absolute-value inequalities
Since absolute value measures distance from zero, an absolute-value inequality becomes a compound inequality. Less-than means “close to zero,” an and: |x| < 5 turns into -5 < x < 5. Greater-than means “far from zero,” an or: |x| > 5 turns into x < -5 or x > 5. A handy mnemonic is “less-thand and great-or.”
Solve |2x - 1| <= 7 (less-than -> 'and') Rewrite as a double inequality: -7 <= 2x - 1 <= 7 Add 1 to all parts: -6 <= 2x <= 8 Divide all parts by 2: -3 <= x <= 4 Interval notation: [-3, 4].