Inequalities Cheat Guide: Linear, Compound, and Absolute Value

Inequalities show up in optimization, engineering tolerances, and almost every real-world constraint problem ("the budget must not exceed…"). The mechanics are similar to solving equations, with one critical twist: multiplying or dividing by a negative flips the inequality sign. This guide collects every move you need on a single page.

Linear inequalities

Treat them exactly like linear equations — except flip the sign whenever you multiply or divide both sides by a negative.

Solve $-3x + 5 < 14$:

1. Subtract 5: $-3x < 9$.
2. Divide by $-3$ and flip: $x > -3$.

The solution set is the open interval $(-3, \infty)$.

Compound inequalities

A compound inequality combines two simpler ones with and (intersection) or or (union).

Solve $-1 \le 2x - 3 < 5$ (an "and" sandwich):

1. Add 3 across all three parts: $2 \le 2x < 8$.
2. Divide by 2: $1 \le x < 4$.

Solution: $[1, 4)$.

For "or" inequalities like $x < -2$ or $x \ge 5$, the solution is two disjoint pieces: $(-\infty, -2) \cup [5, \infty)$.

Absolute value inequalities

The trick: $|A| < k$ rewrites as $-k < A < k$, while $|A| > k$ rewrites as $A < -k$ or $A > k$.

Solve $|2x - 1| \le 5$:

1. Rewrite: $-5 \le 2x - 1 \le 5$.
2. Add 1: $-4 \le 2x \le 6$.
3. Divide by 2: $-2 \le x \le 3$. Solution $[-2, 3]$.

Quadratic inequalities

Move everything to one side, factor, then test sign on each interval.

Solve $x^2 - x - 6 > 0$:

1. Factor: $(x - 3)(x + 2) > 0$.
2. Roots split the line into three intervals: $(-\infty, -2)$, $(-2, 3)$, $(3, \infty)$.
3. Test a point from each: at $x = -3$ the product is positive; at $x = 0$ negative; at $x = 4$ positive.
4. Solution: $(-\infty, -2) \cup (3, \infty)$.

Common mistakes

• Forgetting to flip when dividing by a negative — the single biggest error.
• Mixing up open and closed brackets: $<$ uses parentheses, $\le$ uses brackets.
• Squaring both sides of $|A| < B$ blindly: only valid when both sides are non-negative.

Verify with the AI Inequality Solver

Type any inequality into the Inequality Solver and you will see the full step list — perfect for double-checking homework.

Related references:

• Equation Solver — same algebra, equality version
• Absolute Value Solver — for $|A| = k$ flavor problems
• Quadratic Solver — paired with the quadratic case above