Sec 7.6 Quadratic Inequalities Learning Objectives: 1. Solve Quadratic Inequalities Using Graphical Method 2. Solve Quadratic Inequalities Using Algebraic Method 1. Solve Quadratic Inequalities Using Graphical Method Definition: Quadratic Inequality An inequality of the form: ax 2 + bx + c > 0 ; ax 2 + bx + c < 0 ; ax 2 + bx + c ≥ 0 ; ax 2 + bx + c ≤ 0 Where a, b, c are real numbers with a ≠ 0 Solutions of Quadratic Inequality y y x x
Example 1. Use the graph of the quadratic function f to determine the solution. y
a. f ( x ) > 0 b. 5 f (x ) = 0 c. f ( x ) ≤ 0 –5
Steps for Solving Quadratic Inequalities using Graphical Method 1. Rewrite the inequality in the standard form that is one side is 0. 5
–5 2. Graph the function f ( x ) = ax 2 + bx + c and label the x‐intercepts and y‐intercept of the graph. 3. From the graph, determine where the function is positive and determine where the function is negative. Use the graph to determine the solution set to the inequality. Example 2. Solve inequality. Graph the solution set and state the solution set. 1. x 2 − 3x − 4 ≥ 0 1 x
2. Solve Quadratic Inequalities Using Algebraic Method Steps for Solving Quadratic Inequalities using Algebraic Method y x
1. Rewrite the inequality in the standard form that is one side is 0. 2. Determine the solutions to the equation ax 2 + bx + c = 0 . 3. Use the solutions to the equation solved in step 2 to separate the real number line into intervals. 4. Write ax 2 + bx + c in factored form. Within each interval formed in step 3, determine the sign of each factor. Then determine the sign of the product. Also determine the value of ax 2 + bx + c at each solution found in step2. • If the product of the factors is positive, then ax 2 + bx + c > 0 for all numbers x in the interval. • If the product of the factors is negative, then ax 2 + bx + c < 0 for all numbers x in the interval. Example 3. Solve using algebraic method. Graph the solution set and state the solution set. 1. − x 2 + 8 > 2 x 2 2. Solve f ( x ) ≥ 1 if f ( x ) = 2 x 2 + 4 x ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Example 4. Find the domain of the given function. 1. 3 x 2 + 2 x − 63