Quadratic and Other Nonlinear Inequalities Section 10.5 Quadratic and Other Nonlinear Inequalities

Objectives Solve quadratic inequalities Solve rational inequalities Graph nonlinear inequalities in two variables

Objective 1: Solve quadratic inequalities A quadratic inequality can be written in one of the standard forms where a, b, and c are real numbers and a ≠ 0. To solve a quadratic inequality in one variable, we will use the following steps to find the values of the variable that make the inequality true. Write the inequality in standard form and solve its related quadratic equation. Locate the solutions (called critical numbers) of the related quadratic equation on a number line. Test each interval on the number line created in step 2 by choosing a test value from the interval and determining whether it satisfies the inequality. The solution set includes the interval(s) whose test value makes the inequality true. Determine whether the endpoints of the intervals are included in the solution set.

EXAMPLE 1 Solve: x2 + x – 6 < 0 Strategy We will solve the related quadratic equation x2 + x – 6 = 0 by factoring to determine the critical numbers. These critical numbers will separate the number line into intervals. Why We can test each interval to see whether numbers in the interval are in the solution set of the inequality.

EXAMPLE 1 Solution Solve: x2 + x – 6 < 0 The expression x2 + x – 6 can be positive, negative, or 0, depending on what value is substituted for x. Solutions of the inequality are x-values that make x2 + x – 6 less than 0. To find them, we will follow the steps for solving quadratic inequalities.

EXAMPLE 1 Solve: x2 + x – 6 < 0 Solution

EXAMPLE 1 Solve: x2 + x – 6 < 0 Solution

EXAMPLE 1 Solve: x2 + x – 6 < 0 Solution

Objective 2: Solve Rational Inequalities Rational inequalities in one variable such as can also be solved using the interval testing method. Solving Rational Inequalities Write the inequality in standard form with a single quotient on the left side and 0 on the right side. Then solve its related rational equation. Set the denominator equal to zero and solve that equation. Locate the solutions (called critical numbers) found in steps 1 and 2 on a number line. Test each interval on the number line created in step 3 by choosing a test value from the interval and determining whether it satisfies the inequality. The solution set includes the interval(s) whose test value makes the inequality true. Determine whether the endpoints of the intervals are included in the solution set. Exclude any values that make the denominator 0.

EXAMPLE 3 Solve: Strategy This rational inequality is not in standard form because it does not have 0 on the right side. We will write it in standard form and solve its related rational equation to find any critical numbers. These critical numbers will separate the number line into intervals. Why We can test each interval to see whether numbers in the interval are in the solution set of the inequality.

EXAMPLE 3 Solve: Solution

EXAMPLE 3 Solve: Solution

EXAMPLE 3 Solve: Solution

EXAMPLE 3 Solve: Solution

Objective 3: Graph Nonlinear Inequalities in Two Variables Graphing Inequalities in Two Variables Graph the related equation to find the boundary line of the region. If the inequality allows equality (the symbol is either ≤ or ≥ ), draw the boundary as a solid line. If equality is not allowed (< or >), draw the boundary as a dashed line. Pick a test point that is on one side of the boundary line. (Use the origin if possible.) Replace x and y in the original inequality with the coordinates of that point. If the inequality is satisfied, shade the side that contains that point. If the inequality is not satisfied, shade the other side of the boundary. We use the same procedure to graph nonlinear inequalities in two variables.

EXAMPLE 6 Graph: y < −x2 + 4 Strategy We will graph the related equation y = –x2 + 4 to establish a boundary parabola. Then we will determine which side of the boundary parabola represents the solution set of the inequality. Why To graph a nonlinear inequality in two variables means to draw a “picture” of the ordered pairs (x, y) that make the inequality true.

EXAMPLE 6 Solution Graph: y < −x2 + 4 The graph of the boundary y = –x2 + 4 is a parabola opening downward, with vertex at (0, 4) and axis of symmetry x = 0 (the y-axis). Since the inequality contains an < symbol and equality is not allowed, we draw the parabola using a dashed curve. To determine which region to shade, we pick the test point (0, 0) and substitute its coordinates into the inequality. We shade the region containing (0, 0) because its coordinates satisfy y < –x2 + 4 .