Algebra Placement Test Practice

The following multiple choice algebra questions guide you in assessing your skills and abilities in the following topics: substituting values into algebraic expressions, setting up equations for given situations, basic operations such as multiplication and division of polynomials, factoring polynomials, solving linear equations in one variable, exponents and radicals, rational expressions and linear equations in two variables are presented along with their answers and detailed solutions.

Questions for Algebra Placement Practice


  1. If \( a = -2 \) and \( b = -2 \), what is the value of \[ \frac{a^{3} - 1}{b - 1} \ ? \]
    1. 3
    2. -9
    3. -3
    4. 9
    5. 6
    Detailed Solution.

  2. If \[ -2(x + 9) = 20, \] then \(-4x =\)
    1. -76
    2. -19
    3. 0
    4. 76
    5. 90
    Detailed Solution.

  3. \(x\) is a variable such that if 20% of it is added to its fifth, the result is equal to 12 subtracted from seven-tenths of \(x\). Find \(x\).
    1. 0.2
    2. \(\frac{1}{5}\)
    3. \(\frac{7}{10}\)
    4. 20
    5. 40
    Detailed Solution.

  4. The area \(A\) of a trapezoid is given by \[ A = 0.5(b + B)h, \] where \(b\) and \(B\) are the lengths of the bases, and \(h\) is the height. Express \(B\) in terms of \(A\), \(b\), and \(h\).
    1. \(0.5Ah - b\)
    2. \(\frac{2A}{h} - b\)
    3. \(\frac{2A - b}{h}\)
    4. \(\frac{2A}{h} + b\)
    5. \(\frac{2A + b}{h}\)
    Detailed Solution.

  5. Tom, Linda, and Alex have a total of \$120. Alex has one-third of what Tom has, and Linda has twice as much as Alex. How much money does Linda have?
    1. 10
    2. 20
    3. 40
    4. 60
    5. 80
    Detailed Solution.

  6. Which of the following is equivalent to \[ 6x^{2} - 11x - 2 \ ? \]
    1. \((6x - 1)(x + 2)\)
    2. \((3x - 1)(2x + 2)\)
    3. \((3x + 1)(2x - 2)\)
    4. \((6x + 1)(x - 2)\)
    5. \((x + 1)(6x - 2)\)
    Detailed Solution.

  7. Which of the following is a factor of \[ x^{2} - 7x - 8 \ ? \]
    1. \(x + 1\)
    2. \(x + 8\)
    3. \(x + 7\)
    4. \(x - 1\)
    5. \(x - 7\)
    Detailed Solution.

  8. \[ (2xy^{2} - 3x^{2}y) - (2x^{2}y^{2} - 4x^{2}y) = \]
    1. \(2x^{2}y^{2}\)
    2. \(-2x^{2}y^{2} - 7x^{2}y - 2x^{2}y^{2}\)
    3. \(-2x^{2}y^{2} + x^{2}y - 2x^{2}y^{2}\)
    4. \(2x^{2}y^{2} + x^{2}y - 2x^{2}y^{2}\)
    5. \(-2x^{2}y^{2} + x^{2}y + 2xy^{2}\)
    Detailed Solution.

  9. During the same journey, Stuart drove \(x\) miles for 2 hours, and 200 miles for 3 hours. Find \(x\) if the average speed for the entire journey is 70 miles per hour.
    1. 166
    2. 167
    3. 150
    4. 140
    5. 120
    Detailed Solution.

  10. Given the equations of the lines: \[ \text{(I) } 2y + 3x = 3 \] \[ \text{(II) } -3y - 2x = 5 \] \[ \text{(III) } -6y + 4x = 9 \] \[ \text{(IV) } 2y + 6x = 9 \] Which two lines are perpendicular?
    1. (I) and (II)
    2. (II) and (III)
    3. (III) and (IV)
    4. (I) and (III)
    5. (IV) and (II)
    Detailed Solution.

  11. If \(f(x) = (x + 1)^{2}\), then \(f(t + 2) =\)
    1. \(t^{2} + 2t + 4\)
    2. \(t^{2} + 4\)
    3. \(t^{2} + 6t + 9\)
    4. \(t^{2} + 9\)
    5. \(t^{2} + 2t + 1\)
    Detailed Solution.

  12. For \(x > 0\) and \(y > 0\): \[ (\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) - (\sqrt{x} - \sqrt{y})^{2} = \]
    1. \(-2\sqrt{xy} - 2y\)
    2. \(x - y\)
    3. 0
    4. \(-2\sqrt{xy} + 2y\)
    5. \(2\sqrt{xy} - 2y\)
    Detailed Solution.

  13. What is the slope of the line whose equation is: \[ \frac{x}{2} - \frac{y}{4} = 7 \]
    1. 2
    2. \(\frac{1}{2}\)
    3. \(-\frac{1}{4}\)
    4. \(-\frac{1}{2}\)
    5. -2
    Detailed Solution.

  14. For \(x > 3\): \[ \left( \frac{x}{x - 3} + \frac{1}{2} \right) \left( \frac{2}{x - 1} \right) = \]
    1. \(\frac{x + 1}{(x - 3)(x - 1)}\)
    2. \(\frac{3}{x - 3}\)
    3. \(\frac{x + 3}{2(x - 1)}\)
    4. \(\frac{2x}{(x - 3)(x - 1)}\)
    5. \(\frac{1}{x - 3}\)
    Detailed Solution.

  15. In a standard rectangular coordinate system, point \(A\) has coordinates \((2, 1)\). What must be the coordinates of point \(B\) if \(M(3, 2)\) is the midpoint of segment \(AB\)?
    1. (1, 1)
    2. (4, 3)
    3. (3, 4)
    4. (3, 2)
    5. (2, 3)
    Detailed Solution.

Answers to the Above Questions

  1. A
  2. D
  3. E
  4. B
  5. C
  6. D
  7. A
  8. E
  9. C
  10. D
  11. C
  12. E
  13. A
  14. B
  15. B

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