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Intermediate Algebra Appeals Practice Test 6/07
Show all work and answers. Give all answers in simplest form.
2 3
1. Evaluate 3 4 .
3 7
3 1
2. Evaluate 5 2 .
5 2
1 3
3. Evaluate 9 4 .
4 8
4. What is 15.2 percent of 76? Round answer to the nearest tenth.
5. 30 is 20% of what number?
6. 12 is what percent of 108? Round answer to the nearest tenth of a percent.
7. If 2.5 pounds of steak costs $12.50, how much will 4 pounds cost?
8. Evaluate -3 + 2 ∙ -4.
9. Evaluate (-2)2 – 3 ∙ -1.
10. Simplify 3x 5 2 x 4 .
11. Simplify 23x 1 5 x 6 .
3x 4 y 3
12. Simplify .
12 xy 5
13. Determine the common solution: x + y = 4 and 2x – y = 11.
14. Determine the common solution: 3x - 5y = 14 and 2x – 3y = 9.
9
15. Convert to a percent. Do not round your answer.
16
16. Determine l if P 2l w when P 280 feet and w 40 feet.
17. Determine the slope of the line through:
a. (2,-3) and (-1,5)
b. (4,-1) and (4,6)
c. (3,5) and (-3,5)
18. Write in slope-intercept form y mx b the equation of the line through:
a. (-4,-2) and (0,-3)
b. (-3,2) and (4,5)
c. (5,-1) and (2,-1)
19. Determine the slope for:
a. the line y 5
b. the line 2 x 3 y 4
c. the line x 8
Intermediate Algebra Appeals Practice Test 6/07
20. Write the equation of the line graphed below.
6
-6 6
-6
21. Determine (a) the domain and (b) the range of the function graphed below.
6
-6 6
-6
22. Graph the line 3x 4 y 12 .
23. Determine the distance between the points (-5,2) and (1,-5). Round your answer to the hundredths
place.
24. State which of the following relations are functions (there may be more than one correct answer):
a. 2,0, 2,4, 2,8
b. 3,2, 5,6, 5,3, 8,9
c. 1,3, 1,3, 5,2
d. 5,2, 4,6
2
Intermediate Algebra Appeals Practice Test 6/07
25. Let f x 2 4 x . Determine (a) f 0 , (b) f 1 , (c) f 1
26. Let g x 9 2 x 2 . Determine (a) g 0 , (b) g 2 , (c) g(-2)
27. Solve for x by factoring:
a. 2 x 2 9 x 5
b. x 2 6 x 9 0
c. 9 x 2 4 0
28. Solve for x using the quadratic formula:
a. x 2 3x 2 0
b. 2 x 2 4 x 1
c. x 2 2x 5 0
29. Let Gt 8 2 t , represent the number of rabbits in a barn after t weeks. How many rabbits are there
after (a) 1 week? (b) 3 weeks?
30. Simplify each of the following. Express each answer using positive exponents only.
a. x 5 y z 4 x 3 y 2 z 7
b. x 4 y 2 x 4 y 0
c. x 2 y 5 z x 3 y 2 z 2
31. Compute 3 811 2 .
2 x 2 12 x 32 x 2 10 x 16
32. Multiply, simplifying as much as possible: .
x 2 16 x 64 x 2 3x 10
x 2 2 x 24 x 2 7 x 6
33. Divide, simplifying as much as possible: 2 .
x 2 6x 8 x x6
9x 2 7
34. Add, simplifying as much as possible: 2 .
3x 2 x 8 3x x 4
2
3
Intermediate Algebra Appeals Practice Test 6/07
4x 2 2
35. Subtract, simplifying as much as possible: .
x x 20 x 4
2
36. Draw a graph of y x 2 2 x 15 . In your graph, clearly label and give both coordinates of the x-
intercepts, the y-intercept, and the vertex.
37. Compute log 3 27 .
38. Compute log 64 4 .
x2
39. Determine the domain of the function y .
x 2x 8
2
3x
40. Solve for x: 2.
x 1
2x 6 28
41. Solve for x: 2 .
x3 x3 x 9
4
Intermediate Algebra Appeals Practice Test 6/07
Answers/Solutions for Intermediate Algebra Appeals Practice Test
2 3 3 1
1. Evaluate 3 4 . 2. Evaluate 5 2 .
3 7 5 2
To add or subtract fractions as mixed numbers, you must To add or subtract fractions as mixed numbers, you must
find a common denominator for the fractions. The least find a common denominator for the fractions. The least
common denominator for 3 and 7 is 21. common denominator for 5 and 2 is 10.
2 2x7 14 3 3x2 6
3 3 3 5 5 5
3 3x7 21 5 5x2 10
3 3x3 9 1 1x 5 5
4 4 4 2 2 2
7 7x3 21 2 2x5 10
Add the fractions: [Add numerators, keep common Subtract the fractions: [subtract numerators, keep common
denominators] denominators]
14 9 23 2 6 5 1
1
21 21 21 21 10 10 10
Add the whole numbers: 3 + 4 = 7 Subtract the whole numbers: 5 - 2 = 3
2 2
Add the two results: 7 1 8
21 21
2 1
Answer: 3
Answer: 8 10
21
1 3 4. What is 15.2 percent of 76? Round answer to the
3. Evaluate 9 4 . nearest tenth.
4 8
To add or subtract fractions as mixed numbers, you must
find a common denominator for the fractions. The least Translate into a mathematical equation.
common denominator for 4 and 8 is 8. What is 15.2 percent of 76?
1 2 ↓ ↓ ↓ ↓ ↓
9 9 N = 0.152 • 76
4 8
3 3 Multiply 0.152 and 76 to get 11.552
4 4
8 8
Subtract the fractions: You cannot subtract 3 from 2; you 11.552 rounded to the nearest tenth is 11.6.
8 2 10
must ―borrow‖ 1 from 9 [ 1 ] to get 8 1 8
8 8 8
10 3 7
Now, subtract the fractions:
8 8 8
Subtract the whole numbers: 8 - 4 = 4
7
Answer: 4 Answer: 11.6
8
5. 30 is 20% of what number? 6. 12 is what percent of 108? Round answer to the
nearest tenth of a percent.
Translate into a mathematical equation. Translate into a mathematical equation.
30 is 20% of what number? 12 is what percent of 108
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
30 = 0.20 • N or 0.2N = 30 12 = N • 108 or 108N = 12
Solve for N: N = 30 ÷ 0.2 = 150 Solve for N: N = 12 ÷ 108 = 0.1111
Write N as a percent and round to the nearest tenth.
N = 0.1111 = 11.11% = 11.1%
Answer: 150 Answer: 11.1%
7. If 2.5 pounds of steak costs $12.50, how much will 8. Evaluate -3 + 2 ∙ -4.
4 pounds cost?
Let x = the cost of 4 pounds of steak. Order of operations: Multiply first, then add
; 2.5x 412.50 50
2.5 4
12.50 x -3 + 2 ∙-4 = -3 + (-8) = -11
x 50 2.5 20
Answer: The cost of 4 pounds of steak is $20.00 Answer: -11
5
Intermediate Algebra Appeals Practice Test 6/07
9. Evaluate (-2)2 – 3 ∙ -1. 10. Simplify 3x 5 2 x 4 3x 35 2 x 4
4 31 4 3 7 3x 15 2 x 4 3x 2 x 15 4
Answer: 7 Answer: 5x-11
11. Simplify 23x 1 5x 6 . 3x 4 y 3
23x 21 15x 16
12. Simplify
12 xy 5
6 x 2 5x 6 6 x 5x 2 6
3 1 x4 y3 1
; x 41 x 3 ; 5
y 3 5 y 2
12 4 x y y2
1x 3
Answer: x-4 Answer:
4y2
13. Determine the common solution: 14. Determine the common solution: 3x – 5y = 14 and
x + y = 4 and 2x – y = 11. 2x – 3y = 9.
To find the common solution, eliminate one of the To solve for y, multiply the first equation by -2 and the
unknowns by addition or subtraction; then solve for the second equation by 3; then add the equations
remaining unknown 23x 5 y 14 6 x 10 y 28
x y 4 32 x 3 y 9 6 x 9 y 27
2 x y 11 y 1
3x 0 15 Substitute -1 for y in the first equation:
So x = 15 ÷ 3 = 5. Substitute 5 for x in first equation: 5 + y 3x – 5(-1) =14 3x + 5 = 14 3x = 9 x = 3
= 4 y = 4 – 5 = -1 Check in the second equation: 2(3) -3(-1) = 6 + 3 = 9
Answer: (5,-1) Answer: (3,-1)
9 16. Determine l if P 2l w when P 280 feet
15. Convert to a percent. Do not round your
16 and w 40 feet.
answer. P 2l w 280 2l 40 2l 80
To convert a fraction to a percent, divide the numerator by
the denominator; move the decimal point two places to the 2l 80 280 2l 280 80 200
9 2l 200 l 100
right and affix the % sign. 9 16 0.5625 56.25% Answer: l = 100 feet
16
Answer: 56.25%
17. Determine the slope of the line through: 18. Write in slope-intercept form y mx b the
a. (2,-3) and (-1,5) equation of the line through:
b. (4,-1) and (4,6) a. (-4,-2) and (0,-3)
c. (3,5) and (-3,5) b. (-3,2) and (4,5)
The slope between 2 points, (x1, y1) & (x2,y2) is c. (5,-1) and (2,-1)
y y1 3 2 1
m 2 ; if x2 – x1 =0, the slope is undefined a) Find the slope: m
x 2 x1 0 4 4
1
5 3 8 8 Find b: 2 4 b 2 1 b b 2 1 3 .
a) m Answer: 4
1 2 3 3 1
Answer: y x3
4
6 1 7
b) m Undefined Answer: Undefined 52 3
44 b) Find the slope: m
4 3 7
0
9 9 23
Find b: 2 3 b 2
55 3
c) m
0
0 Answer: 0 b b 2 .
33 6 7 7 7 7
3 23
Answer: y x
7 7
1 1 0
c) Find the slope: m 0
25 3
Find b: 1 0 5 b b 1
Answer: y 0x 1 or y 1
6
Intermediate Algebra Appeals Practice Test 6/07
19. Determine the slope for: 20. Write the equation of the line graphed below
a. the line y 5
6
b. the line 2x 3 y 4
c. the line x 8
Write each equation in slope-intercept form: y mx b , if
possible.
a) y 0 x 5 m 0 , Answer: Slope is 0.
2 4 2
b) 2 x 3 y 4 3 y 2 x 4 y x m -6 6
3 3 3
2
Answer: Slope is
3
c) x = 8 cannot be written in slope-intercept form. The
slope is undefined
Answer: Slope is undefined
-6
The x-intercept is (-1,0); the y-intercept is (0,-2).
2 0 2
The slope is 2
0 1 1
Ans: The equation of the line is y 2x 2
21. Determine (a) the domain and (b) the range of the 22. Graph the line 3x 4 y 12
function graphed below Let x = 0; then 0 – 4y = -12 and y = 3
6 Let y = 0; then 3x – 0 = -12 and x = -4
-6 6
-6
The domain of the function f is the set of all input values
for which the function is defined. The range of a function
is the set of all output values for the given function.
Looking at the graph we see that the input values, x, for this
function are x 5 and the out put values, y, are y 3
Ans: Domain: x x 5
Range: y y 3
7
Intermediate Algebra Appeals Practice Test 6/07
23. Determine the distance between the points (-5,2) 24. State which of the following relations are
and (1,-5). Round your answer to the hundredths functions (there may be more than one correct
place. answer):
A function is a relation in which each x-coordinate is
D x2 x1 2 y 2 y1 2 paired with one and only one y-coordinate.
For the relations given, only c and d are FUNCTIONS
So,
D 1 52 5 22 62 72 since they are the only sets in which every x is paired with
a unique y-value.
36 49 85 9.2195 a) is NOT a function because -2 is paired with three
different values of y. b) is NOT a function because 5 is
paired with two different values of y.
Answer: 9.22 Answer: c) and d) are functions
25. Let f x 2 4 x . 26. Let g x 9 2 x 2
Determine (a) f 0 , (b) f 1 , (c) f 1 Determine (a) g 0 , (b) g 2 , (c) g 2
a) f 0 2 40 2 0 2 Answer: f 0 2
a) g 0 9 202 9 0 9 Answer: g 0 9
b) f 1 2 41 2 4 2 Answer: f 1 2
b) g 2 9 22 9 8 1 Answer: g 2 1
2
c) f 1 2 41 2 4 6 Answer: f 1 6
c) g 2 9 2 2 9 8 1 Answer: g 2 1
2
27. Solve for x by factoring: 28. Solve for x using the quadratic formula:
a. 2 x 2 9 x 5 b. x 2 6 x 9 0 c. 9 x 2 4 0 a. x 2 3x 2 0 b. 2 x 2 4 x 1 c. x 2 2 x 5 0
a) Put equation in ax 2 bx c 0 form: 2 x 2 9 x 5 0 The quadratic formula gives solution for equations in the
To factor 2 x 2 9 x 5 0 , you must find two numbers form ax 2 bx c 0 .
whose sum is -9 [b] and whose product is -10 [ac = 2(-5)]. b b 2 4ac
The numbers, -10 and 1, are used as coefficients of x and x Quadratic Formula
2a
substituted for -9x to get 2x 2 10x 1x 5 , a polynomial
with four terms. Now, we factor 2x from the first two a) x 2 3x 2 0; a 1, b 3, c 2
terms and 1 from the third and fourth terms to get:
2 xx 5 1x 5 x 52 x 1 b 2 4ac 32 41 2 9 8 9 8 17
To solve 2 x 2 9 x 5 x 52 x 1 0 , set each factor 17 cannot be simplified.
equal to 0 and solve for x: x 5 0 x 5 and 3 17 3 17
x
2 x 1 0 2 x 1 x
1
Answer: 5 and
1 21 2
2 2
3 17
Answer: x
2
b) To factor x 2 6 x 9 0 , you must find two numbers
whose sum is -6 and whose product is +9. The numbers are
b) 2 x 2 4 x 1 2 x 2 4 x 1 0; a 2, b 4, c 1
-3 and -3.
x 2 6 x 9 x 2 3x 3x 9 xx 3 3x 3 b 2 4ac 42 421 16 8 8
x 3x 3 x 32 8 can be simplified to 4 2 2 2
x 2 6 x 9 x 32 0 x 3 0 x 3 x
42 2 42 2 2 2
Answer: x=3 22 4 2
2 2
Answer: x
c) 9 x 2 4 3 x 2 is a special product—the
2 2
2
difference of two squares. Its factors are (3x+2) and (3x-2) c) x 2 x 5 0; a 1, b 2, c 5
2
2
9 x 2 4 3x 23x 2 3x 2 0 3x 2 x b 2 4ac 22 415 4 20 16
3
2 16 can be simplified to 4i .
and 3x 2 0 3x 2 x
3 2 4i 2 4i
x 1 2i
2 2 21 2
Answer: x ,
3 3 Answer: x 1 2i
8
Intermediate Algebra Appeals Practice Test 6/07
29. Let Gt 8 2 t , represent the number of rabbits 30. Simplify each of the following. Express each
answer using positive exponents only.
in a barn after t weeks. How many rabbits are
there after (a) 1 week? (b) 3 weeks?
5
a) x y z 4 x 3 y 2 z 7 x 53 y 1 2 z 4 7
a) t 1 g 1 8 21 16 y3z3
x -8 y 3 z 3 Answer:
Answer: There are 16 rabbits after 1 week. x8
b) t 3 g 3 8 2 3 64 b) x 4 y 2 x 4 y 0 x 44 y 20 x 0 y 2 1 y 2
Answer: There are 64 rabbits after 3 weeks.
Answer: y 2
c) x 2 y 5 z x 3 y 2 z 2 x 23 y 52 z 12
x 1 y 3 z 1
xy 3
Answer:
z
31. Compute 3 811 2 32. Multiply, simplifying as much as possible:
3 81
12
3 81 3 9 27 2 x 12x 32 x 2 10x 16 2x 2x 8 x 2x 8
2
x 2 16x 64 x 2 3x 10 x 8x 8 x 2x 5
Ans: 27 2x 2
Ans:
x 5
33. Divide, simplifying as much as possible: 34. Add, simplifying as much as possible:
2 2 2 2
x 2 x 24 x 7 x 6 x 2 x 24 x x 6
9x 2 7 9x 2 7
2 2 2 2
x 6x 8 x x6 x 6x 8 x 7x 6 3x 2 x 8
2
3x x 4
2 3x 4x 2 3x 4x 1
x 6 x 4 x 3x 2 x3
LCD = 3x 4x 2x 1
x 4 x 2 x 6 x 1 x 1
9 x 2x 1 7 x 2
x 3
Ans:
x 1
3x 4 x 2 x 1 3x 4 x 2 x 1
2
9x 7x 2 7 x 14
3x 4x 2x 1 3x 4x 2x 1
9 x 7 x 2 7 x 14
2 2
9 x 16
3x 4 x 2 x 1 3x 4x 2x 1
3 x 43 x 4 3 x 4
3x 4x 2x 1 x 2x 1
9
Intermediate Algebra Appeals Practice Test 6/07
35. Subtract, simplifying as much as possible: 36. Draw a graph of y x 2 2 x 15 . In your graph,
4x 2 2 4x 2 2
clearly label and give both coordinates of the x-
x x 20 x 4 x 5x 4 x 4
2
intercepts, the y-intercept, and the vertex
LCD = x 5x 4 y 15 x 2 2 x y 15 1 x 2 2 x 1
y 16 x 12
4x 2 2x 5 4 x 2 2 x 10
Vertex‖ (1,-16); y-int: (0,-15); x-int: (-3,0) and (5,0)
x 5x 4 x 4x 5 x 5x 4
2x 8 2x 4 2
x 5x 4 x 5x 4 x 5
(-3,0) (5,0)
(0,-15) (1,-16)
37. Compute log 3 27 38. Compute log 64 4 .
log 3 27 x is equivalent to the exponential equation log 64 4 x is equivalent to the exponential equation
3 x 27 . Since 3 3 27, x 3 1
64 x 4 . Since 3
64 4, x
Ans: log 3 27 3 3
Ans: log 64 4
1
3
39. Determine the domain of the function 3x
40. Solve for x: 2.
x2 x 1
y 2 .
x 2x 8
The domain of function is the set of values of x for which LCD = x 1 , So multiply both sides by the LCD.
the function is defined. Since this is a rational function,
you must look at the denominator x 2 2 x 8 and x 13x
2x 1 3x 2 x 2 x 2
determine the values for which the denominator is zero. x 1
These are the values that must be excluded from the 32 6
domain of the function. The function is not defined for Check: 2
2 1 3
these values since they make the denominator zero.
Ans: x = 2
x 2 2 x 8 0 x 4x 2 0 x 4 or x 2
Ans: Domain: x x 4 and x 2
28 28
LCD = x 3x 3
2x 6 2x 6
41. Solve for x:
x 3 x 3 x2 9 x 3 x 3 x 3x 3
So multiply both sides (all terms) by the LCD:
x 3x 3 2 x x 3x 3 6 x 3x 3 28
x 3 x3 x 3x 3
2 xx 3 6x 3 28 2 x 2 6 x 6 x 18 28 2 x 2 12x 10 0 2x 5x 1 0 x 5 or x 1
25 6 28 10 6 28 10 24 28 14 28 7 7
x 5 :
5 3 5 3 52 9 8 2 25 9 8 8 16 8 16 4 4
Check:
2 1 6 28 2 6 28 1 6 28 7 7
x 1 :
1 3 1 3 1 9
2 4 2 1 9 2 2 8 2 2
Ans: x 5 or x 1
10
