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Chapter 10 Quadratic Equations and Functions

Chapter 10 Quadratic Equations and Functions. Section 6 Quadratic Inequalities. Section 10.6 Objectives. 1 Solve Quadratic Inequalities. Solving Quadratic Inequalities. A quadratic inequality is an inequality of the form ax 2 + bx + c > 0 or ax 2 + bx + c < 0 or

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Chapter 10 Quadratic Equations and Functions

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  1. Chapter 10 Quadratic Equations and Functions Section 6 Quadratic Inequalities

  2. Section 10.6 Objectives 1 Solve Quadratic Inequalities

  3. Solving Quadratic Inequalities A quadratic inequality is an inequality of the form ax2 + bx + c > 0 or ax2 + bx + c < 0 or ax2 + bx + c 0 or ax2 + bx + c 0 where a  0. Solving Quadratic Inequalities (Graphical Method) Step 1: Write the inequality so that ax2 + bx + c is on one side of the inequality and 0 is on the other. Step 2: Graph the function f(x) = ax2 + bx + c. Be sure to label the x-intercepts of the graph. Step 3: From the graph, determine where the function is above the x-axis and determine where the function is below the x-axis. Use the graph to determine the solution set to the inequality.

  4. f(x) < 0 for – 7 < x < 4 y 10 (4, 0) (– 7, 0) 5 x f(x) > 0 for x < – 7 f(x) > 0 for x > 4 8 8 6 4 2 2 4 6 10 15 20 25 (– 1.5, – 30.25) (0, – 28) 30 Solving a Quadratic Inequality Graphically Example: Solve the inequality using the graphical method: x2 + 3x – 28  0 Graph the function f(x) = x2 + 3x – 28 The solution is {x| x  – 7 or x  4}.

  5. Solving Quadratic Inequalities Solving Quadratic Inequalities (Algebraic Method) Step 1: Write the inequality so that ax2 + bx + c is on one side of the inequality and 0 is on the other. Step 2: Determine the solutions to the equation ax2 + bx + c = 0. Step 3: Use the solutions to the equation solved in Step 2 to separate the real number line into intervals. Step 4: Write ax2 + bx + c in factored form. Within each interval formed, determine the sign of each factor, the sign of the product, and the value of each solution. (a) If the product of the factors is positive, then ax2 + bx + c > 0 for all numbers x in the interval. (b) If the product of the factors is negative, then ax2 + bx + c < 0 for all numbers x in the interval. If the inequality is not strict ( or ), include the solutions of ax2 + bx + c = 0 in the solution set.

  6. -8 -7 0 -6 1 -5 2 3 -4 4 -3 5 -2 6 -1 (– 7, 4 ) (–, – 7) (4, ) Solving a Quadratic Inequality Algebraically Example: Solve the inequality. x2 + 3x – 28  0 x2 + 3x – 28 = 0 Solve the equation (x + 7)(x – 4) = 0 Factor. Solve for x. Separate the number line into the following intervals: Continued.

  7. Solving a Quadratic Inequality Algebraically Example continued: The solution set is {x| x – 7 or x  4}.

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