ACT Math Solving and Graphing Inequalities: Interpret Inequality Solutions on Number Lines

Solve inequalities like equations, but flip the sign if you multiply or divide by a negative. Example: 2x>6 becomes x>3 (divide by positive 2, sign stays). Example: -2x>6 becomes x<-3 (divide by negative 2, flip sign). The sign-flip rule is the most critical step. Forgetting it causes wrong answers. Graphing: For x>3, place an open circle at 3 and shade right. For x≥3, use a closed circle at 3 and shade right. Open circle means "not included"; closed circle means "included".

More complex: -2x+5<11. Subtract 5: -2x<6. Divide by -2 (flip): x>-3. Graph: closed circle at -3, shade right.

Mistake 1: Forgetting to flip the sign when dividing by negative. Most errors come from this. Always ask after each operation: "Did I divide or multiply by negative?" If yes, flip. Mistake 2: Confusing open and closed circles. Open means >, <. Closed means ≥, ≤. Mistake 3: Shading the wrong direction. For x>3, shade right (numbers greater than 3 are to the right). For x<3, shade left. Test one point: If x>3, is x=4 a solution? Yes, so 4 should be in the shaded region. This test catches shading errors.

During practice, solve, then verify by testing a point in the shaded region. If it satisfies the inequality, shading is correct.

Problem 1: 3x+2<14. Subtract 2: 3x<12. Divide by 3: x<4. Graph: open circle at 4, shade left. Test: x=3 works. 3(3)+2=11<14 ✓. Problem 2: -x+5≥2. Subtract 5: -x≥-3. Divide by -1 (flip): x≤3. Graph: closed circle at 3, shade left. Problem 3: 2x-7>5. Add 7: 2x>12. Divide by 2: x>6. Graph: open circle at 6, shade right. Problem 4: -3x≤9. Divide by -3 (flip): x≥-3. Graph: closed circle at -3, shade right. Problem 5: 4x+1<9. Subtract 1: 4x<8. Divide by 4: x<2. Graph: open circle at 2, shade left. Solve and graph each, testing one point to verify shading direction.

Find 10 inequality problems from a practice test. For each, solve, graph, and test a point. By the tenth problem, inequality solving and graphing will be automatic.

Inequality questions appear on most ACT Math tests. They combine algebraic solving with visual graphing. Students who master both solving and graphing pick up 1-2 points because the combination tests two skills that many students don't fully master.

Drill inequalities daily this week. Each day, solve and graph five inequalities, testing points to verify correctness. By test day, you should solve and graph any inequality within 60 seconds.