ACT Math Quadratic Inequalities: Solve Parabola-Based Inequality Problems

A quadratic inequality like x²-3x+2>0 asks where the parabola is above or below the x-axis. Method: (1) Find zeros by solving x²-3x+2=0. Factor: (x-1)(x-2)=0, so x=1 or x=2. (2) Graph the parabola. Since the coefficient of x² is positive, it opens upward. (3) For >, find where parabola is above x-axis: x<1 or x>2. For <, find where it's below x-axis: 1Visualization makes the answer obvious: sketch the parabola, mark zeros, shade the regions satisfying the inequality.

Another example: -x²+4<0 (parabola opens downward, below x-axis means the inequality is satisfied). Zeros: -x²+4=0, x=±2. Downward parabola below x-axis when x<-2 or x>2.

Mistake 1: Forgetting to check the parabola direction (opens up or down). This determines whether you want regions above or below the x-axis. Mistake 2: Finding zeros correctly but graphing the regions wrong. Always sketch the parabola and shade the correct side. Mistake 3: Confusing inequality symbols. > means above x-axis (or the shaded region for downward parabola). < means below x-axis. Visual graphing prevents these errors better than pure algebra.

During practice, always sketch the parabola, mark zeros, and shade regions. This visual approach is faster and more reliable than memorizing rules.

Problem 1: x²-5x+6>0. Zeros: (x-2)(x-3)=0, so x=2, 3. Parabola opens up. Above x-axis when x<2 or x>3. Problem 2: -x²+9≤0. Zeros: x=±3. Parabola opens down. Below x-axis when x≤-3 or x≥3. Problem 3: x²-4<0. Zeros: x=±2. Parabola opens up. Below x-axis when -20. Zeros: -(x²-2x-3)=0, (x-3)(x+1)=0, x=3,-1. Parabola opens down. Above x-axis when -1 Solve and graph each, shading the correct region.

Find five quadratic inequality problems from a practice test. For each, find zeros, sketch the parabola, shade regions, and write the solution. By the fifth problem, quadratic inequality solving will feel intuitive through graphing.