ACT Math Compound Inequalities: Solve in Your Head With One Key Rule

Compound inequalities like -2Key rule: Reverse the inequality sign only if you multiply or divide all parts by a negative number. If you add, subtract, multiply by positive, or divide by positive, the sign stays the same. This single rule trips up many students, so lock it in: sign flips only with negative multiplication/division.

Example showing the rule: If -2x<6, divide all parts by -2 (negative number), which reverses: x>-3. If 2x<6, divide by 2 (positive number), which keeps the sign: x<3. This distinction makes all the difference between a correct and incorrect answer.

Mistake 1: Forgetting to reverse the sign when multiplying or dividing by a negative. This is the most common error. Always ask after each operation: "Did I multiply or divide by negative?" and reverse if yes. Mistake 2: Reversing the sign when you shouldn't. If you add or subtract, never reverse. Mistake 3: Setting up a three-part ("sandwich") inequality incorrectly. For -2Always work systematically from outside to inside when solving compound inequalities.

Create a visual reminder card: write "FLIP = Negative × or ÷" and tape it to your desk this week. This reinforces the critical rule.

Problem 1: -112. Divide by -4 (flip sign): x<-3. Problem 5: -2≤3x-5<4. Add 5: 3≤3x<9. Divide by 3: 1≤x<3. Problem 6: 6≥-2x≥-8. Divide by -2 (flip signs): -3≤x≤4. Solve all six, showing each step and marking where you reverse the sign (if at all).

Find five compound inequality questions from a practice test. Solve them step-by-step, pausing after each operation to ask "Did I multiply or divide by negative?" By the fifth question, the method will feel automatic.

Compound inequalities appear on most ACT Math tests, usually in questions 30-45. Once you know the sign-flip rule, these problems are mechanical. Students who master compound inequalities pick up 1-2 points because the skill is straightforward and rewards careful execution of one simple rule.

Drill compound inequalities daily this week. Each day, solve five problems, focusing on identifying where to reverse the sign. By test day, you should solve any compound inequality in under 60 seconds.