INTEGRATED MATH 3 UNIT 2: INEQUALITIES AND LINEAR PROGRAMMING

1. INTEGRATED MATH 3UNIT 2: INEQUALITIES AND LINEAR PROGRAMMING INEQUALITIES IN ONE VARIABLE LESSON OBJECTIVES: Write inequalities to express questions about functions of one or two variables.

2. Given a graph of one or more functions, solve inequalities related to the function(s). • Solve quadratic inequalities in one variable by solving the corresponding equation algebraically and reasoning about the graph of the related function(s).

3. Describe the solution set of an inequality in one variable symbolically, as a graph on a number line and using interval notation.

4. In earlier courses, you learned how to solve a variety of problems by representing and reasoning about them with Algebraic equations and inequalities. Let’s consider a raffle fundraising situation as shown with the worksheet.

5. Suppose that plans for the raffle show profit P is predicted to depend on the ticket price x by the function: P(x) = -2,500 + 5,000x – 750x2 Let’s consider questions important to the fundraising raffle that can be answered by solving inequalities involving this profit function.

6. What would you learn from solutions of the following inequalities? • -2,500 + 5,000x – 750x2 >0 • P(x) < 0 • -2,500 + 5,000x – 750x2 ≥4,000 • P(x) ≤ 2,500

7. The fundraiser is predicted to give a positive profit because in the graph, the parabola or curve is upward. • The fundraiser is predicted to lose money because the profit is shown to be less than 0.

8. The fundraiser is predicted to raise at lease $4,000 in profit because the equation is shown to be greater than or equal to $4,000. • The fundraiser is predicted to raise at most $2,500 because the equation shows the profit to be less than or equal to $2,500.

9. How could you use the graph to estimate the solutions of the inequalities in Part a? • Above the x-axis. • Below the x-axis. • On or above the line y=4,000 • On or below the line y=2,500

10. In what ways could you record solutions of the inequalities in words, symbols or diagrams? • In words: when x is greater than $0.50 and less than about $6.10. • As a number line graph: 0 .5 1 2 3 4 5 6 7 8

11. Symbolically: 0.5 < x < 6.1 • Using interval notation introduced in investigation 3: (0.5, 6.1)

12. The height of a main support cable of a suspension bridge comes from the following equation: h(x) = 0.04x2– 3.5x + 100 where x is horizontal distance from the left end of the bridge and h(x) is the height of the cable above the bridge surface.

13. Where is the bridge cable less than 40 feet above the bridge surface? • Where is the bridge cable at least 60 feet above the bridge surface?

14. How far is the cable above the bridge surface at a point 45 feet from the left end? • Where is the cable 80 feet above the bridge surface?

15. 0.04x2 – 3.5x + 100 < 40 23.4 < x < 64.1 • 0.04x2– 3.5x + 100 ≥ 60 x ≤ 13.5 or x ≥ 74

16. In this case, a number line graph does not really make any sense because what is asked is the value of h(x) corresponding to the given x = 45. 0.04(45)2 - 3.5(45) +100 = 23.5

17. This has two solutions. 0.04x2 – 3.5x + 100 = 80 x ≈ 6.1 and x ≈ 81.4

18. LET’S DO SOME EXTRA PROBLEMS Describe the solutions of these inequalities using symbols and number line graphs: • 7t – t2 < 0 • h2 + 2h - 3 ≤ 0 • -d2 – 12d – 20 > 0 • X2 + 0.5x – 3 > 0

19. ANSWERS: • t < 0 or t > 7 • -3 ≤ h ≤ 1 • 10 < d < -2 • x < -2 or x > 1.5

20. MORE PROBLEMS! Describe the solutions of these inequalities using symbols and number line graphs: • 7t – t2 < 10 • 5h2+ 14h < 3 • -d2 – 11d – 20 > 4 • -X2+ 6x – 8 < 1

21. ANSWERS: • t < 2 or t > 5 • -3 < h < 0.2 • -8 < d < -3 • x < 3 or x > 3

22. HOMEWORK!! Complete the homework page and quiz. You are allowed to work in groups for the homework. The quiz is closed book and closed notes. Have a great day and make sure you take your time!!