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 below.

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

    Solution

    Substitute \( x = -2 \) into \( f(x) \):

    \( f(-2) = 4(-2)^3 - 4(-2)^2 + 10 = -32 - 16 + 10 = -38 \)

  2. Which value of \( x \) satisfies \( -7x + 6 \leq -8 \)?

    Solution

    Solve the inequality:

    \( -7x + 6 \leq -8 \Rightarrow -7x \leq -14 \Rightarrow x \geq 2 \)

    Answer: D

  3. What is the domain of \( f(x) = \sqrt{6 - 2x} \)?

    Solution

    Expression inside the radical must be non-negative:

    \( 6 - 2x \geq 0 \Rightarrow x \leq 3 \)

    Domain: \( x \leq 3 \)

  4. Are the lines \( y = 2x \) and \( 2y = -x \) parallel, perpendicular, or neither?

    Solution

    Convert \( 2y = -x \) to \( y = -\frac{1}{2}x \).

    Slopes: 2 and \( -\frac{1}{2} \); product is -1 ⇒ perpendicular.

    Answer: B

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

    Solution

    Rewrite: \( |-2x - 5| = k + 3 \)

    No solution if \( k + 3 \lt 0 \Rightarrow k \lt -3 \)

    Answer: A

  6. Translate the statement: "The price is no less than 100 Dollars."

    Solution

    "No less than" means "greater than or equal to":

    \( x \geq 100 \)

  7. Which relation does not represent a function?

    Solution

    In choice C, \( x = 2 \) maps to both 3 and 7 ⇒ not a function.

    Answer: C

  8. Which point does not lie on \( y = -x + 3 \)?

    Solution

    Only (2,2) fails: \( 2 \neq -2 + 3 = 1 \)

    Answer: D

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

    Solution

    Let \( m \) be the new slope: \( m(-5) = -1 \Rightarrow m = \frac{1}{5} \)

    Answer: C

  10. Which property justifies \( 3(xy) = (3x)y \)?

    Solution

    This is the Associative Property of Multiplication.

    Answer: D

  11. Where do the lines \( x = 3 \) and \( y = -4 \) intersect?

    Solution

    They intersect at (3, -4) ⇒ Quadrant IV.

    Answer: C

  12. Evaluate: \( 2^{-|-2|} \)

    Solution

    \( 2^{-2} = \frac{1}{4} = 0.25 \)

    Answer: B

  13. If \( a, b \) are positive, evaluate \( (a^0 - 3b^0)^5 \)

    Solution

    \( (1 - 3)^5 = (-2)^5 = -32 \)

    Answer: C

  14. Translate: "Length \( L \) is at most 45 cm"

    Solution

    \( L \leq 45 \)

    Answer: D

  15. The equation \( mx - 8 = 6 - 7(x + 3) \) has no solution if:

    Solution

    Simplify: \( x(m + 7) = -7 \Rightarrow m \neq -7 \)

    Answer: C

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

    Solution

    Both sides simplify to \( -mx + 1 = -4x + 1 \Rightarrow m = 4 \)

    Answer: B

  17. Which statement is always true?

    Solution

    Every function is a relation.

    Answer: C

  18. Which inequality has no solution?

    Solution

    \( |x + 3| \lt -2 \) has no solution (absolute value is always ≥ 0)

    Answer: B

  19. Lines \( y = (a - 5)x + 5 \) and \( y = -2x + 7 \) are parallel if:

    Solution

    Match slopes: \( a - 5 = -2 \Rightarrow a = 3 \)

    Answer: A

  20. Lines \( y = (a - 5)x + 5 \) and \( y = -2x + 7 \) are perpendicular if:

    Solution

    Slopes multiplied equal -1: \( -2(a - 5) = -1 \Rightarrow a = \frac{11}{2} \)

    Answer: D

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