Intermediate Algebra Practice Questions and Answers
Sample Set 5

Problem: If \( f(x) = 4x^3 - 4x^2 + 10 \), then \( f(-2) = \)

1. 26
2. -38
3. 10
4. 38

Solution:

Substitute \( x = -2 \) into the function: \[ f(-2) = 4(-2)^3 - 4(-2)^2 + 10 = 4(-8) - 4(4) + 10 = -32 - 16 + 10 = -38 \] Correct Answer: B (-38)

Problem: Which of these values of \( x \) satisfies the inequality \( -7x + 6 \leq -8 \)?

1. -2
2. 0
3. -7
4. 2

Solution:

Solve the inequality for \( x \): \[ -7x + 6 \leq -8 \implies -7x \leq -14 \implies x \geq 2 \] Among the choices, only 2 satisfies \( x \geq 2 \).
Correct Answer: D (2)

Problem: The domain of the function \( f(x) = \sqrt{6 - 2x} \) is given by:

1. \( x > 0 \)
2. \( x \geq 6 \)
3. \( x \leq 3 \)
4. \( x \leq 6 \)

Solution:

The expression under the square root must be greater than or equal to zero: \[ 6 - 2x \geq 0 \implies -2x \geq -6 \implies x \leq 3 \] Correct Answer: C (\( x \leq 3 \))

Problem: The lines \( y = 2x \) and \( 2y = -x \) are:

1. parallel
2. perpendicular
3. horizontal
4. vertical

Solution:

The slope of the first line is \( m_1 = 2 \).
Rewrite the second line in slope-intercept form: \( 2y = -x \implies y = -\dfrac{1}{2}x \), so its slope is \( m_2 = -\dfrac{1}{2} \).
Since the product of their slopes is \( 2 \times \left(-\dfrac{1}{2}\right) = -1 \), the lines are perpendicular.
Correct Answer: B (perpendicular)

Problem: The equation \( |-2x - 5| - 3 = k \) has no solution if \( k = \)

1. -5
2. -3
3. 7
4. 0

Solution:

Isolate the absolute value expression: \[ |-2x - 5| = k + 3 \] Since an absolute value is always non-negative (\( \geq 0 \)), the equation has no solution when the right-hand side is negative: \[ k + 3 < 0 \implies k < -3 \] Among the options, only -5 satisfies this condition.
Correct Answer: A (-5)

Problem: The inequality corresponding to the statement: "the price \(x \) is no less than 100 Dollars" is:

1. \( x < 100 \)
2. \( x \geq 100 \)
3. \( x \leq 100 \)
4. \( x > 100 \)

Solution:

"No less than" means greater than or equal to (\( \geq \)).
Correct Answer: B (\( x \geq 100 \))

Problem: Which of these relations DOES NOT represent a function?

1. \( \{(2, 3), (-4, 3), (7, 3)\} \)
2. \( \{(0, 0), (-1, -1), (2, 2)\} \)
3. \( \{(2, 3), (-5, 3), (2, 7)\} \)
4. \( \{(-1, 3), (-5, 3), (-9, 0)\} \)

Solution:

A relation is not a function if a single input (\( x \)-value) is paired with more than one output (\( y \)-value). In choice C, the input \( 2 \) is paired with both \( 3 \) and \( 7 \).
Correct Answer: C

Problem: Which of these points DOES NOT lie on the graph of \( y = -x + 3 \)?

1. (9, 6)
2. (3, 0)
3. (-2, 5)
4. (2, 2)

Solution:

Test point (2, 2): \[ y = -(2) + 3 = 1 \neq 2 \] Since the equation evaluates to 1 when \( x = 2 \), the point (2, 2) does not lie on the line.
Correct Answer: D (2, 2)

Problem: What is the slope of the line perpendicular to the line \( y = -5x + 9 \)?

1. 5
2. -5
3. \( \dfrac{1}{5} \)
4. \( -\dfrac{1}{5} \)

Solution:

The slope of the given line is \( -5 \). The perpendicular slope is the negative reciprocal: \[ m_{\perp} = -\dfrac{1}{-5} = \dfrac{1}{5} \] Correct Answer: C (\( \dfrac{1}{5} \))

Problem: Which property is used to write: \( 3(xy) = (3x)y \)?

1. Commutative property of multiplication
2. Multiplicative inverse property
3. Distributive property
4. Associative property of multiplication

Solution:

Changing the grouping of factors in multiplication without changing their order is the associative property of multiplication.
Correct Answer: D (Associative property of multiplication)

Problem: In which quadrant do the lines \( x = 3 \) and \( y = -4 \) intersect?

1. I
2. II
3. III
4. IV

Solution:

The lines intersect at the point \( (3, -4) \). Since the \( x \)-coordinate is positive and the \( y \)-coordinate is negative, the point lies in Quadrant IV.
Correct Answer: C (IV)

Problem: The value of \( 2^{-|-2|} \) is:

1. 4
2. 0.25
3. -4
4. -0.25

Solution:

Evaluate the absolute value exponent: \[ |-2| = 2 \implies 2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4} = 0.25 \] Correct Answer: B (0.25)

Problem: If \( a \) and \( b \) are positive real numbers, then \( (a^0 - 3b^0)^5 = \)

1. 0
2. 1
3. -32
4. 32

Solution:

Since any non-zero base to the power of 0 equals 1 (\( a^0 = 1 \) and \( b^0 = 1 \)): \[ (1 - 3(1))^5 = (1 - 3)^5 = (-2)^5 = -32 \] Correct Answer: C (-32)

Problem: Which inequality describes the situation: "length \( L \) is at most 45 cm"?

1. \( L = 45 \text{ cm} \)
2. \( L > 45 \text{ cm} \)
3. \( L \geq 45 \text{ cm} \)
4. \( L \leq 45 \text{ cm} \)

Solution:

"At most" means less than or equal to (\( \leq \)).
Correct Answer: D (\( L \leq 45 \text{ cm} \))

Problem: The equation \( mx - 8 = 6 - 7(x + 3) \) DOES NOT have any solution if \( m = \)

1. 3
2. 7
3. -7
4. 0

Solution:

Simplify both sides of the equation: \[ mx - 8 = 6 - 7x - 21 \] \[ mx - 8 = -7x - 15 \] Rearrange to group terms with \( x \): \[ mx + 7x = -7 \implies x(m + 7) = -7 \] For the equation to have no solution, the coefficient of \( x \) must be zero while the constant on the right is non-zero: \[ m + 7 = 0 \implies m = -7 \] Correct Answer: C (-7)

Problem: The equation \( -mx + 1 = 13 - 4(x + 3) \) is an identity if \( m = \)

1. 4
2. -4
3. 1
4. -1

Solution:

Expand the right-hand side: \[ -mx + 1 = 13 - 4x - 12 \] \[ -mx + 1 = -4x + 1 \] For this equation to be an identity (true for all \( x \)), the coefficients of \( x \) must match: \[ -m = -4 \implies m = 4 \] Correct Answer: A (4)

Problem: Which of the following is ALWAYS true?

1. A function is not a relation
2. Every function is a relation
3. Every relation is a function
4. A relation is not a function

Solution:

By definition, a function is a specific type of relation where each input maps to a single output. Therefore, every function is a relation.
Correct Answer: B (Every function is a relation)

Problem: Which of these inequalities has NO solutions?

1. \( 7x + 4 > 3 \)
2. \( |x - 1| > -10 \)
3. \( |x + 3| < -2 \)
4. \( -100x + 5 \geq 8 \)

Solution:

An absolute value is always greater than or equal to zero (\( \geq 0 \)). It can never be strictly less than a negative number like \(-2\). Thus, \( |x + 3| < -2 \) has no solution.
Correct Answer: C (\( |x + 3| < -2 \))

Problem: The lines \( y = (a - 5)x + 5 \) and \( y = -2x + 7 \) are parallel if \( a = \)

1. -2
2. 3
3. 5
4. -5

Solution:

Parallel lines have identical slopes. Set the slopes equal to each other: \[ a - 5 = -2 \implies a = 3 \] Correct Answer: B (3)

Problem: The lines \( y = (a - 5)x + 5 \) and \( y = -2x + 7 \) are perpendicular if \( a = \)

1. \( \dfrac{11}{2} \)
2. 5
3. \( -\dfrac{2}{9} \)
4. \( \dfrac{9}{2} \)

Solution:

Perpendicular lines have slopes whose product is \(-1\): \[ (a - 5)(-2) = -1 \implies a - 5 = \dfrac{1}{2} \implies a = 5.5 = \dfrac{11}{2} \] Correct Answer: A (\( \dfrac{11}{2} \))