# Math questions/algebra Essay

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7.4 Exercises

8. In number (8), the correct half-plane that will complete the graph is the plane above the line y=3. The graph of y>3 is

y>3

To know which plane to shade, we will have to consider test points. If we consider (0,0), x=0 and y=0, but 0 is not greater than 3, so the plane containing (0,0) is not the correct half plane. If we consider (1,5), x=1, y=5 and 5>3. Thus, the correct plane is the one that contains (1,5) or the plane above the line.

16. 4x + y ≥ 4. First, graph the line 4x+y=4 by solving for the intercepts. If x=0, y=4. Hence, the y-intercept is 4. If y=0, then the x-intercept is 1. Then, we’ll have to get a test point. If we consider (0,0), and put the values 0 to x and y, then 4(0) + 0 = 0 which is not greater than 4, so the correct half-plane is the one that does not contain (0,0), so the graph must look like this:

28. Just like number 16, graph the line 3x – 4y = 12 using intercepts. If x=0, y= -3, and if y=0, x=4. Taking (0,0) as a test point, we will have 3(0) – 4(0) = 0 which is less than 12. Hence, the half-plane that contains (0,0) is the correct plane. The graph is:

38. Let x be the number of dimes and y be the number of quarters. The inequality, since a dime is worth .10 and a quarter is worth .25, is .10x + .25 ≥ 30.

7.5 Exercises

2. f(x) = x2 – 7x + 10, find f(0), f(5), f(-2). By substitution of the values to x, (a) f(0) = 02 -7(0) + 10 = 10. (b) f(5) = 52 – 7(5) + 10 = 25 – 35 + 10 = 0 and (c) f(-2) = (-2)2 – 7(-2) +10 = 4 + 14 + 10 = 28.

18. -3x+4y = 11 → 4y = 11 + 3x → y = ( 11+3x) / 4 or y = 11/4 + 3x/4

22. In graphing functions, set f(x) as y and graph the line through intercepts. That is, if x=0, then the value of y = -2(0) – 5 = -5. Also, if y=0, 0 = -2x – 5 → 2x = -5 → x = -5/2. Thus, (0,-5) and (-5/2, 0) is in the graph of f(x). The graph:

30. f(x) = 4x – 3, by substituting -1 to x, f(-1) = 4(-1) – 3 = -4 – 3 = -7.

36. f(x) = 5x – 1, by substituting (a – 2) to x, f(a -2) = 5(a – 2) -1= 5a – 10 – 1 = 5a – 11.

54. The profit function is given by P(x) = 2.25x – 7000. (a) If he sells 2000 units, then x= 2000. P(2000) = 2.25(2000) – 7000 = 4500 -7000 = -2500. Since P(2000) < 0, then he will not earn any profit. (b) P(5000) = 2.25(5000) – 7000 = 4250. He will earn 4250 in selling 5000 units. (c). The break even point is when P(x) = 0, thus 0 = 2.25x – 7000, which implies that x is approximately 3112 units.

References

Linear Functions. (n.d). Retrieved June 26, 2007, from http://id.mind.net/~zona/mmts/ ~functionInstitute/ linearFunctions/linearFunctions.html

Linear Functions. (n.d). Retrieved June 26, 2007, from http://www.shodor.org/ ~interactivate/discussions/LinearFunctions/