Q&A for How to Solve Quadratic Inequalities

When you divide or multiply an inequality by a negative number, you need to flip the inequality sign.

Here's the easiest way to solve for x: Let a = (x² + 1). Then a² + 2a - 35 = 0. By factoring, (a + 7)(a - 5) = 0. Solving for a, a = -7 or 5. Then (x² + 1) = -7 or 5. If (x² + 1) = -7, x² = -8, and x = +/-√-8 = +/-2i√2 (both "imaginary" numbers). If (x² +1) = 5, x² = 4, and x = +/- 2, (which are "real" numbers). If you prefer, you may reject the imaginary roots, leaving x = +/- 2. Both of the "real" roots work when plugged back into the original equation.

(x + 2) / (x + 1) > 4. Multiply both sides by (x + 1): (x + 2) > 4(x + 1). Then (x + 2) > 4x + 4. Subtract x on both sides: 2 > 3x + 4. Now subtract 4 on both sides: -2 > 3x. Then divide both sides by 3: -2/3 > x.

Start by expanding and simplifying the inequality to x^2 - 3x - 18 > 0. Then factor it to (x + 3)(x - 6) > 0, and solve for the intervals where this expression is true. The solutions are x > 6 or x < -3.