EXAMPLE 6 Use a quadratic inequality as a model The number T of teams that have participated in a robot- building competition for high school students can be modeled by Robotics T(x) = 7.51x 2 –16.4x , 0 ≤ x ≤ 9 Where x is the number of years since For what years was the number of teams greater than 100?

EXAMPLE 6 Use a quadratic inequality as a model T(x) > x 2 – 16.4x > x 2 – 16.4x – 65 > 0 Graph y = 7.51x 2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9. The graph’s x -intercept is about 4.2. The graph lies above the x -axis when 4.2 < x ≤ 9. There were more than 100 teams participating in the years 1997–2001. ANSWER SOLUTION You want to find the values of x for which:

EXAMPLE 7 Solve a quadratic inequality algebraically Solve x 2 – 2x > 15 algebraically. SOLUTION First, write and solve the equation obtained by replacing > with =. x 2 – 2x = 15 x 2 – 2x – 15 = 0 (x + 3)(x – 5) = 0 x = – 3 or x = 5 Write equation that corresponds to original inequality. Write in standard form. Factor. Zero product property

EXAMPLE 7 Solve a quadratic inequality algebraically The numbers – 3 and 5 are the critical x -values of the inequality x 2 – 2x > 15. Plot – 3 and 5 on a number line, using open dots because the values do not satisfy the inequality. The critical x -values partition the number line into three intervals. Test an x -value in each interval to see if it satisfies the inequality. Test x = – 4: Test x = 1: 1 2 –2(1) 5 –1 >15 Test x = 6: The solution is x 5. ANSWER (– 4) 2 –2(– 4) = 24 >15  6 2 –2(6) = 24 >15 

GUIDED PRACTICE for Examples 6 and 7 6. Robotics Use the information in Example 6 to determine in what years at least 200 teams participated in the robot-building competition. SOLUTION You want to find the values of x for which: T(x) > x 2 – 16.4x > x 2 – 16.4x – 165 > 0

GUIDED PRACTICE for Examples 6 and 7 Graph y = 7.51x 2 – 16.4x – 165 on the domain 0 ≤ x ≤ 9. There were more than 200 teams participating in the years 1998 – ANSWER

GUIDED PRACTICE for Examples 6 and 7 7. Solve the inequality 2x 2 – 7x = 4 algebraically. SOLUTION First, write and solve the equation obtained by replacing > with 5. Write equation that corresponds to original inequality. Write in standard form. Factor. Zero product property 2x 2 – 7x = 4 2x 2 – 7x – 4 = 0 (2x + 1)(x – 4) = 0 x = – 0.5 or x = 4

GUIDED PRACTICE for Examples 6 and 7 The numbers 4 and – 0.5 are the critical x -values of the inequality 2x 2 – 7x > 4. Plot 4 and – 0.5 on a number line, using open dots because the values do not satisfy the inequality. The critical x -values partition the number line into three intervals. Test an x -value in each interval to see if it satisfies the inequality. Test x = – 3 : Test x = 2 : Test x = 5 : – 3 – 4 – 2 – – 5 – 6 – 7 2 (– 3) 2 – 7 (– 3) > 4  2 (5) 2 – 7 (3) > 4  2 (2) 2 – 7 (2) > 4 The solution is x 4. ANSWER