Download Ch 7 Sections 7

Ch 7 (Sections 7.4 – 7.5) and 8 Cumulative Test
Name: __________________________ Date: _____________
1. Graph the inequality.
2x + 3y > 6
2
First we write the equation as 3 y  2 x  6  y   x  2
3
2
So we first graph the line y   x  2 . The y-intercept is (0, 2). The x-intercept is
3
(3, 0).
y
2
x
0
3
The inequality is the shaded area.
2. Given g(x) = –3x + 5, find g(2a).
The y-intercept is (0, 5). The x-intercept is 0=-3x+5, so x = 5/3.
g(x)
5
x
0
5/3
g (2a)  3* (2a)  5  6a  5
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3. Graph the inequality.
x – y  2   y  x  2  y  x  2
y-intercept is (0, -2). x-intercept is (2, 0).
First graph the line y=x-2. The inequality is the shaded area including the line.
y
x
2
-2
4. Graph the inequality.
y  3x
The line y=3x passes through (0,0) and (1, 3). It is the shaded area including the line.
y
x
0
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20
5. Given f(x) = –x3 – 3x2 – 3x + 9, find f(–2), f(0), and f(3)
f(x)
11
9
0
0
-2
3
1
x
-20
-40
-54
-60
-80
f (2)  (2)3  3  (2) 2  3  (2)  9  11
f (0)  9
f (3)  33  3  32  3  3  9  54
-100
-3
-2
-1
0
1
2
3
4
6. Given f(x) = –5x – 1, find f(–2).
The y-intercept is (0, -1). 0= -5x -1, then 5x =-1. so x= -1/5. The x-intercept is (-1/5, 0).
f (2)  5  (2)  1  9
y
-1/5
0
x
-1
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7. Graph f(x) = 4x + 1.
The y-intercept is (0, 1). 4x+1=0, so x =-1/4. So the x-intercept is (-1/4, 0).
y
1
-1/4
0
x
8. Graph f(x) = –2x + 4.
x-intercept is (2, 0). Y-intercept is (0, 4).
y
4
2
0
x
9. Solve the system by addition.
5x – 3y = 13
(1)
4x – 3y = 11
(2)
(1)-(2): x =2.
Plug x =2 into (1): 5*2-3y=13, -3y=3, y= -1.
So the solution is x = 2, y = -1.
10. Solve the system by substitution.
x + y = 12
y = 2x
Substitute y=2x into the first equation: x + 2x = 12, 3x = 12, so x = 4.
Then y = 2*4 = 8
Solution: x =4, y =8.
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11. Adult tickets for a play cost $20 and child tickets cost $12. If there were 23 people at a
performance and the theater collected $348 from ticket sales, how many adults and how
many children attended the play?
Assume there are x adults and y children.
x + y = 23, so y = 23 –x.
and 20x + 12y = 348. Substitute y = 23 –x to the above equation,
20x + 12(23 – x) = 348
8x = 348 – 12 *23 = 72, so x = 9.
Then y = 23 – 9 = 14.
So there are 9 adults and 14 children.
12. A home-based company produces both hand-knitted scarves and sweaters. The scarves
take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is
limited to 40 hours per week, and the total production capacity is 5 items per week.
Write a system of inequalities representing this situation, where x is the number of
scarves and y is the number of sweaters. Then graph the system of inequalities.
It takes 2x hours to produce x scarves, and 14 hours to produce y sweaters. The total
maximum labor hours is 40 *5 =200
So 2 x  14 y  200 , and x  0, y  0
The x and y-intercepts are (100, 0) and (0, 100/7) respectively for the line 2x + 14y =
200.
y
100/7
0
100
x
13. The sum of two numbers is 34. Their difference is 12. What are the two numbers?
Assume the two numbers are x and y.
x + y = 34
x – y = 12
Add the two equation, 2x = 46, x = 23
Then y = 34 - x = 34 - 23 = 11.
So the two numbers are 11 and 23.
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14. Solve the system by substitution.
3x – 5y = 15
y = 2x + 11
Substitute y into the first equation
3x  5(2 x  11)  15  3x  10 x  55  15  7 x  70  x  10
y  2 x  11  2  (10)  11  9
Solution: x = -10, y = -9.
15. Solve the system by substitution.
3x – y = –7
x + y = –9
From x + y =-9, we have y = -x – 9. Substitute into the first equation
3x  ( x  9)  7  3x  x  9  7  4 x  16  x  4
Then y = -(-4) – 9 = -5.
Solution: x = -4, y = -5.
16. The sum of two numbers is 49. The second is 5 more than 3 times the first. What are
the two numbers?
Assume the first number x, the second is y, then y = 3x + 5.
(1) y –x = 49. Substitute y, we have 3x + 5 –x = 49, then 2x = 44, so x = 22.
y = 3 * 22 + 5 = 71
(2) x – y = 49, so x –( 3x + 5) = 49. -2x - 5 = 49, -2x = 54, x = -27
Y = 3* (-27) + 5 = -76
So the numbers are 22 and 77, or -27 and -76.
17. The difference of two numbers is 75. The second is 5 less than 5 times the first. What
are the two numbers?
Assume the first number is x, the second is y, then y = 5x – 5.
(1) y –x = 75, then 5x – 5 – x = 75, 4x = 80, x =20
So y = 5 * 20 – 5 = 95.
(2) x- y = 75, then x – (5x -5) = 75, -4x = 70, x = -17.5
So y = 5 * (-17.5) -5 = -92.5.
So the two numbers are 20 and 95, or -17.5 and -92.5.
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18. Solve the following system of linear inequalities by graphing.
x – 3y > 6
3x + 2y > 12
The line x -3y = 6: x-intercept: (6, 0), y-intercept (0, -2).
Line 3x +2y =12: x-intercept: (4, 0), y-intercept: (0, 6)
The blue line is 3x + 2y =12.
The system is the shaded area excluding the lines.
y
6
6
x
4
0
-2
19. Given f(x) = x2 + x + 10, find f(0).
f ( x  0)  0  0  10  10
20. Rewrite the equation y = –2x + 5 as a function of x.
1
5
2x   y  5  x   x 
2
2
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