Quadratic Inequalities in One Variable Quadratic inequalities in one variable can be written in one of the following forms: a b c + + 0 a b c + + 0 a b c + + 0 a b c + + 0 Where a, b, and c are real and a 0. When solving inequalities we are trying to find all possible values of the variable which will make the inequality true. Quadratic inequalities in one variable can be solved algebraically or graphically. METHOD 1: CASE ANALYSIS A quadratic inequality can be solved by factoring the quadratic epression and then using case analysis. The product of the two factors will be negative when the factors have different signs. The product of the two factors will be positive when the factors have the same signs. There are two possible cases in each of these situations. Eample 1: Solve a Quadratic Inequality in One Variable by Case Analysis Solve the following inequalities by using case analysis. a. 3 0 b. 10 c. + + 1 0 Solutions: a. 3 0 First, write the inequality in factored form. Then, consider the possible cases. The product of the two factors will be negative when the factors have different signs. There are two ways for this to happen. Case 1: The first factor is positive and the second factor is negative. Case : The first factor is negative and the second factor is positive.
b. 10 First, write the inequality in standard form and factor. Then, consider the possible cases. The product of the two factors will be positive when the factors have the same signs. There are two ways for this to happen. Case 1: Both factors are positive. Case : Both factors are negative. c. + + 1 0 First, multiply the inequality by 1 so the coefficient of the quadratic term is positive. Write the inequality in factored form. Consider the possible cases. Case 1: Case :
METHOD : ROOTS & TEST POINTS A quadratic inequality can be solved by finding the roots of the related equation and marking these roots (known as the critical values) on a number line, which separate the line into intervals. Then, choose a test point in each interval and evaluate the quadratic epression for that value. Write a plus or minus sign over that interval on the number line to indicate whether the value of the epression is positive or negative and use this information to solve the inequality. Eample : Solve a Quadratic Inequality in One Variable by Using Roots and Test Points Solve the following inequalities by using roots and test points. a. 15 0 b. 5+ 6 0 c. 7 1 Solutions: a. 15 0 First solve the related quadratic equation 15 = 0 to find the roots, or critical values. Then, plot the critical values on the number line. Use closed circles since these values are solutions to the inequality. The critical values divide the number line into three intervals: 3, 3 5 and 5. Choose one test point from each interval. Substitute each value into the quadratic inequality to determine whether the value satisfies the inequality. Use the table below to organize your results. Interval 3 3 5 5 Test Point Substitution 15 Is 15 0?
b. + 5 6 0 First solve the related quadratic equation 5+ 6 = 0 to find the roots, or critical values. Then, plot the critical values on the number line. Use open circles since these values are not solutions to the inequality. Complete the table below to determine the solution to the inequality. Interval Test Point Substitution Is + 5 6 0? c. 7 1 First solve the related quadratic equation 7 1 = to find the roots, or critical values. Then, plot the approimate roots on the number line. Use circles. Complete the table below to determine the solution to the inequality. Interval Test Point Substitution Is 7 1 0?
Eample 3: Apply Solving a Quadratic Inequality in One Variable The inequality 4.9t + 15t+ 0 models the time, t, in seconds that a baseball is in flight. Determine the time interval that the baseball is in flight. Solution: METHOD 3: GRAPH THE CORRESPONDING FUNCTION A quadratic inequality can be solved using the graph of the corresponding quadratic function by visually identifying the part of the graph that satisfies the given condition. If the quadratic is greater than zero, then the solutions will be the -values for which the graph is above the -ais. If the quadratic is less than zero, then the solutions will be the -values for which the graph is below the -ais. Eample 4: Solve a Quadratic Inequality in One Variable by Using a Graph Solve the following inequalities by using the graph of the corresponding function. a. 6 0 b. + 6 8 0 c. + + 1 0 d. + + 1 0 e. + 10 5 0 Solutions: a. 6 0 Determine the roots of the related quadratic equation 6= 0. Plot the roots on a number line and draw a sketch of the related quadratic function Note: The roots of the related equation are the -intercepts of the function. y = 6. 6 0, identify the part of the graph that is below the -ais.
b. + 6 8 0 Determine the roots of the related quadratic equation + 6 8 = 0. Plot the roots on a number line and draw a sketch of the related quadratic function y = + 6 8. + 6 8 0, identify the part of the graph that is on or below the -ais. c. + + 1 0 Determine the roots of the related quadratic equation + + 1= 0. Sketch a graph of the related quadratic function y = + + 1. Note: Since the roots of the related equation are non-real, then the graph will have no -intercepts. + + 1 0, identify the part of the graph that is above the -ais. d. + + 1 0 See the previous eample for the graph of the corresponding function y = + + 1. + + 1 0, identify the part of the graph that is below the -ais. Note: The solution could also be stated as "the empty set", represented by the character "Ø".
e. + 10 5 0 Determine the roots of the related quadratic equation + 10 5 = 0. Plot the roots on a number line and draw a sketch of the related quadratic function Note: The roots of the related equation are the -intercepts of the function. y = + 10 5. + 10 5 0, identify the part of the graph that is below the -ais. Eample 5: Apply Solving a Quadratic Inequality in One Variable A medical office has a rectangular parking lot that measures 10 ft by 00 ft. The owner wants to epand the size of the parking lot by adding an equal distance to two sides as shown. If zoning restrictions limit the total size of the parking lot to 35 000 ft, what possible distances can be added? Model this situation with an inequality and solve. Clearly show your method. Interpret your solution to the inequality in the given contet.
Quadratic Inequalities in One Variable Etra Questions Set up and solve an inequality for each of the following situations. Make sure to clearly show your method. Interpret your solution to the inequality in the given contet. 1. A business offers tours to the Amazon. The profit, P, that the company earns for number of tourists can be modeled by p( ) = 5 + 1000 3000. How many people are needed for a profit of at least $5000?. For a driver aged years, a study found that the driver s reaction time V(), in milliseconds, to a visual stimulus such as a traffic light can be modeled by V ( ) = 0.005 0.3 +, 16 70. At what ages does a driver s reaction time tend to be greater than 5 milliseconds? 3. A rectangular construction site is to be enclosed with fencing. The length of the site is 15 m longer than the width. What dimensions are possible if at least 700 m of land is enclosed? Write an inequality to represent this situation, then solve the problem.