# Solutions and Explanations to Intermediate Algebra Questions in Sample 5

Detailed solutions and full explanations to the multiple choice intermediate algebra questions in sample 5 are presented.

1. If f(x) = 4x3 - 4x2 + 10, then f(-2) =
Solution
Substitute x by -2 in f(x) as follows
f(-2) = 4(-2)3 - 4(-2)2 + 10
= 4(-8) - 4(4) + 10 = - 32 - 16 + 10 = - 38
2. Which of these values of x satisfies the inequality -7x + 6 ≤ -8
 A. -2 B. 0 C. -7 D. 2

Solution
Solve the inequality
-7x + 6 ≤ -8 , given
-7x + 6 - 6 ≤ -8 - 6 , add - 6 to both sides
-7x ≤ - 14 , simplify
-7x / -7 ≥ -14 / -7 , divide by - 7 and CHANGE symbol of inequality
x ≥ 2 , solution set
The answer to the above question is D since 2.
3. The domain of the function f(x) = √(6 - 2x) is given by
Solution
f(x) is real if the expression under the radical is positive or equal to zero. Hence to find the domain of we need to solve the following inequality.
(6 - 2x) ≥ 0
x ≤ 3 , domain of f
4. The lines y = 2x and 2y = - x are
 A. parallel B. perpendicular C. horizontal D. vertical

Solution
Horizontal lines are of the form y = constant and vertical lines are of the from x = constant and therefore the two lines are neither horizontal nor vertical. Let us find the slopes of the two given lines
y = 2x has a slope equal to 2
2y = - x is equivalent to y = -(1/2) x and its slope is equal to -(1/2)
Since the slopes are not equal, the two lines are not parallel. The product of the two slopes is given by
2(-1/2) = - 1
and hence the two lines are perpendicular.
5. The equation |-2x - 5| - 3 = k has no solution if k =
 A. -5 B. -3 C. 7 D. 0

Solution
We first rewrite the given equation in the form
|-2x - 5| = k + 3
The term |-2x - 5| is either positive or equal to zero. Therefore the above equation has no solutions whenever the expression k + 3 is negative. The values of k for which the above equation has no solutions are solutions of the inequality
k + 3 < 0 or k < - 3
The answer is A since - 5 is less than - 3.
6. The inequality corresponding to the statement:"the price is no less than 100 Dollars" is
Solution
If the price is no less than 100 Dollars, then the price is either equal to or greater than 100 Dollars.
x ≥ 100
7. Which of these relations DOES NOT represent a function?
 A. {(2,3),(-4,3),(7,3)} B. {(0,0),(-1,-1),(2,2)} C. {(2,3),(-5,3),(2,7)} D. {(-1,3),(-5,3),(-9,0)}

Solution
For the relation in C, when x = 2, there are two possible values of y: 3 or 7 and therefore the relation in C is not a function.
8. Which of these points DOES NOT lie on the graph of y = -x + 3?
 A. (9,- 6) B. (3,0) C. (-2,5) D. (2,2)

Solution
Substitute the coordinates of the given points in the given equation and check which one gives a false statement.
Point (9,- 6) : - 6 = -(9) + 3 , - 6 = - 6 , true , point lies on the line
Point (3,0) : 0 = - (3) + 3 , 0 = 0 , true , point lies on the line
Point (-2,5) : 5 = - (-2) + 3 , 5 = 5 , true , point lies on the line
Point (2,2) : 2 = - (2) + 3 , 2 = 1 , false , point DOES NOT lie on the line
9. What is the slope of the line perpendicular to the line y = -5x + 9?
Solution
The slope of the given (in slope intercept form) line is equal to - 5. Let m be the slope of the line perpendicular to the given line. Two lines are perpendicular if the product of their slopes is equal to -1. Hence
m*(-5) = - 1
Solve for m. Hence
m = 1/5 is the slope of a line perpendicular to the given line.
10. Which property is used to write:3(x y) = (3 x)y?
 A. Commutative property of multiplication B. Multiplicative inverse property C. Distributive property D. Associative property of multiplication

Solution
We may use the Associative property of multiplication to write
3(x y) = (3 x)y
11. In which quadrant do the lines x = 3 and y = - 4 intersect?
Solution
The two lines intersect at the point (3 , -4) which is in quadrant IV.
12. The value of 2 - | - 2 | is
Solution
2 - | - 2 | = 2 - 2 , since | - 2 | = 2
= 1 / 2 2 , since a -n = 1 / an
= 1/4 = 0.25
13. If a and b are positive real numbers, then (a0 - 3b0)5 =
Solution
Simplify.
(a0 - 3b0)5 = (1 - 3*1)5 = (- 2)5 = - 32
14. Which inequality describes the situation:"length L is at most 45 cm".
Solution
C
15. The equation m x - 8 = 6 - 7(x + 3) DOES NOT have any solution if m =
Solution
Solve for x.
m x - 8 = 6 - 7(x + 3)
m x + 7x = 6 - 21 + 8
x(m + 7) = -7
x = - 7 / (m + 7)
m cannot be equal to - 7 otherwise the denominator will be zero.
16. The equation - m x + 1 = 13 - 4(x + 3) is an identity if m =

Solution
Expand both the right side.
- m x + 1 = - 4 x + 1
The above is an identity if m = 4.
17. Which of the following is ALWAYS true?
Solution
Every function is a relation
18. Which of these inequalities has NO solutions?
Solution
The absolute value of any expression is positive or equal to zero. Hence the inequality .
|x + 3| < -2
has no solutions
19. The lines y = (a - 5)x + 5 and y = -2x + 7 are parallel if a =
Solution
Two lines are parallel if their slopes are equal. Hence
a - 5 = - 2
Solve for a
a = 3
20. The lines y = (a - 5)x + 5 and y = -2 x + 7 are perpendicular if a =
Solution
Two lines are perpendicular if the product of their slopes is equal to -1. Hence
-2(a - 5) = - 1
Solve for a
a = 11/2

1. B
2. D
3. C
4. B
5. A
6. B
7. C
8. D
9. C
10. D
11. C
12. B
13. C
14. D
15. C
16. A
17. B
18. C
19. B
20. A

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