Inequalities Explained: Linear, Compound, Quadratic

Inequalities look identical to equations until you reach the rule that flips you up at midnight: when you multiply or divide by a negative number, the inequality direction flips. This guide walks through linear, compound, and quadratic inequalities with the patterns that solve 95% of homework.

The one rule everyone forgets

For equations: every operation preserves the equality. $5 = 5$ implies $5 \cdot (-1) = 5 \cdot (-1)$ — both sides equally negated, equality holds.

For inequalities: multiplying or dividing both sides by a negative number flips the direction. $5 > 3$ is true, but multiply both by $-1$ and we get $-5 > -3$, which is false. The corrected statement is $-5 < -3$.

This single rule is the source of most inequality mistakes. Burn it into your reflexes:

• Add/subtract anything → no flip.
• Multiply/divide by positive → no flip.
• Multiply/divide by negative → flip the inequality.

Linear inequalities

Solve like you solve linear equations, watching out for sign flips.

Example 1: $3x + 5 > 14$.

• Subtract 5: $3x > 9$.
• Divide by $3$ (positive, no flip): $x > 3$.
• Solution set: $(3, \infty)$ — open parenthesis means $x = 3$ is not included.

Example 2 (with the flip): $-2x + 7 \leq 1$.

• Subtract 7: $-2x \leq -6$.
• Divide by $-2$ (negative — FLIP): $x \geq 3$.
• Solution set: $[3, \infty)$ — square bracket because of $\leq$, including $3$.

Compound inequalities

A "compound" inequality joins two simple inequalities with AND or OR.

AND is often written as a single chain: $-1 < 2x + 3 \leq 7$. Operate on all three parts simultaneously.

• Subtract 3 everywhere: $-4 < 2x \leq 4$.
• Divide by 2 everywhere: $-2 < x \leq 2$.
• Solution: $(-2, 2]$.

OR stays as two separate inequalities. The solution is the union of both individual solution sets:

$x < -3$ or $x > 5$ → solution $(-\infty, -3) \cup (5, \infty)$.

Quadratic inequalities

For $x^2 + bx + c > 0$ (or any inequality $\neq 0$):

1. Find roots of $x^2 + bx + c = 0$.
2. Plot roots on the number line — they divide it into intervals.
3. Test a point in each interval to see whether the quadratic is positive or negative there.
4. Pick the intervals matching the inequality direction.

Example: $x^2 - 5x + 6 > 0$.

• Factor: $(x - 2)(x - 3) > 0$. Roots at $x = 2$ and $x = 3$.
• Test intervals:
  • $x = 0$: $(0-2)(0-3) = 6 > 0$ ✓
  • $x = 2.5$: $(0.5)(-0.5) = -0.25 < 0$ ✗
  • $x = 4$: $(2)(1) = 2 > 0$ ✓
• Solution: $(-\infty, 2) \cup (3, \infty)$.

For $\leq$ or $\geq$ inequalities, include the roots (closed intervals): $(-\infty, 2] \cup [3, \infty)$.

Graphing solutions on a number line

• Open circle (○) at a value not included ($<$ or $>$).
• Closed circle (●) at a value included ($\leq$ or $\geq$).
• Arrow extending to infinity in the direction of the solution.

Compound AND → bracket between two circles. Compound OR → two separate rays going outward.

Inequalities with absolute value

$|x - a| < b$ unpacks to $-b < x - a < b$, i.e. $a - b < x < a + b$ — a bounded interval.

$|x - a| > b$ unpacks to $x - a < -b$ OR $x - a > b$, i.e. $x < a - b$ OR $x > a + b$ — two rays going outward.

Common mistakes

1. Forgetting to flip when dividing by a negative. The single biggest source of wrong inequality answers.
2. Including endpoints incorrectly. $<$ vs $\leq$ matters — your bracket type depends on it.
3. Treating compound AND like equals. $-2 < x < 5$ is a single statement; you can't break it into "$x = -2$ or $x = 5$."
4. Solving quadratic inequalities like equations. Setting $x^2 - 4 > 0$ "equal to zero" gives roots $\pm 2$; the inequality solution isn't $\{-2, 2\}$ but the intervals between/around them.

Try it yourself

Drop any inequality (linear, compound, quadratic, with absolute value) into our free Inequality Solver — the AI flips signs correctly and shows every step, plus a number-line solution graph.

Related material:

• Glossary: Inequality
• Glossary: Absolute Value
• Quadratic Equation Calculator — pair with the inequality version