MCC Math Department

Intermediate Algebra Review Questions

(for testing out of MAT 120/122 and INTO MAT 150,151, or 152)

The following is a guide to topics that should be reviewed prior to taking the math placement test. It does not merely target information that is on the placement test but will point the student to areas that are heavily emphasized in intermediate algebra. Additional review material (in the form of software tutorials) is available in the Information Commons to currently enrolled MCC students.

NOTE: While completion of this review material will be helpful in preparing for the math placement test the MCC math department is unable to guarantee desired placement.

As you prepare to take the math placement test keep the following points in mind:

  1. At the present time, calculators are NOT allowed when taking the Asset test.
  2. Since the asset test is a timed multiple choice test study this material with a mind toward speed and accuracy as well as content.
  3. The problems on the asset test are not graduated in difficulty therefore if you get stuck on one go on to another problem.
  4. There are a number of different forms of each asset test therefore this study guide will contain information that may not appear on your particular version of the test.
  5. Questions may be worded differently than in your previous classes. Answers given here may appear in a different form than the one you reach; you need to be able to determine if the two answers are equivalent.

Review Questions

  1. In a given set of numbers identify which are integers, rational, irrational and imaginary. Given:Answers
    1. List the Integers.

    2. List the rational numbers.

    3. List the irrational numbers.

    4. List the imaginary numbers.

  2. Simplify the following.
    3. Answers
    1. |10| - |-3|

    2. -|8 - 6|

    3. |-2 - 3| + |12 + 11|

    4. 5-1 + 5

    5. 8 + 80 - 8-1

  3. Add, subtract, multiply and divide rational expressions.
    4. Answers
    1. Reduce to lowest terms:

    2. Multiply:

    3. Divide:

    4. Add:

    5. Subtract:

    6. Simplify:

    7. Simplify:

  4. Use the laws of exponents to simplify the following expressions. Write your final answers without negative exponents. Assume all variable represent positive values.
    5. Answers
    a. e)
    b.
    c. f)
    d.

  5. Add, subtract, multiply and divide polynomials.
    6. Answers

    1. Add:

    2. Subtract: from

    3. Multiply:

    4. Expand:

    5. Divide: from

  6. Factor by removing GCF's, by grouping, difference of squares, sum and differences of cubes, and trinomials:
    7. Answers

    1. 12km3 - 24k3m2 + 36k2m4 - 60k4m3 e) 125a3 + 343p3

    2. x3y2 + x3 - 3y2 - 3 f) 8m2 - 14mp - 39p2

    3. 16m4 - 1 g) 2p4 + 31p2q2 - 16q4

    4. (x - y)2 - 2(x - y) - 3

  7. Write radicals in simplest form. Assume all variables represent positive numbers.
    8. Answers

  8. Perform the operations on radicals (index 2 or 3) indicated below.
    9. Answers
    1. Rationalize the denominator and simplify:

    2. Multiply and Simplify:

    3. Divide and Simplify: (x, y > 0)

    4. Combine: 5x + 3 - xy

    5. Multiply and Simplify:
    6. Subtract and express answer with rationalized denominator:
    7. Rationalize the denominator:

    8. Multiply and Simplify:

  9. Perform operations on complex numbers indicated below.
    10. Answers

    1. Simplify:

    2. Combine: (3 + 2i) + (-4 - i) - (2 + 5i)

    3. Multiply: (1 + 3i)(2 - 6i)

    4. Find the quotient:

  10. Perform the operations on functions indicated below.
    11. Answers

    1. If f(x) = 3 + 2x and g(x) = x2 - 1 then find:

      1. f(a - 1) v) (fg)

      2. vi)

      3. (f + g)(-1) vii) (f o g)(1)

      4. (f - g)(2) viii) (g o f)(-1)

    2. Find the inverse of a linear function. That is, given f(x) = 3x - 2, find f-1(x).

  11. Use the properties of logarithms as indicated below.
    12. Answers

    1. Write as a single logarithm

      2. i) ii)

    2. Use the properties of logarithms to simplify as a sum, or difference of logarithms

      3. i) ii)

    3. Simplify the following:

      4. i) ii)

    NOTE: Because the placement exam is a multiple choice exam the solutions to equations are often 'disguised'. Instead of responding with the actual solutions to an equation you may be asked to give the sum or product of the solutions.

  12. Solve each of the following equations.

    13. In problems (a) - (c) give the product of the solutionsAnswers.

    1. |x| = 5

    2. |x + 5| = 6

    3. |2x + 4| = |3x - 2|

    In problems (d) - (f) give the solutionsAnswers

    1. 3 - 2x ( x + 1) = x (1 - 2x)

    2. 5x + 10 = 5 ( x + 1)

    3. -3x + 2 ( x + 1) = 2 - x

    In problems (g) - (i) give the sum of the solutionsAnswers

    1. 2x2 + 7x - 13 = 0

    2. 2x(x + 4) = 10

    3. 3x(x-4) = 10&

  13. Solve each of the following equations.
    14. Answers

    )
    a. g)
    b. h)
    c. i)
    d. j)
    e.
    f. (obtain x to the nearest hundredth

  14. Solve the following literal equations for the indicated variable.
    15. Answers

    1. P = 2L + 2W for L

    2. 7x - 2y - 14 = 0 for Y

    3. A = P + PRT for P

  15. Solve word problems of the following types. Ratio, Proportion, Mixture, Geometry, work, I=PRT, money and numerical.

    1. (Numerical) Three less than twice the third of three consecutive integers equals the sum of the three integers. Find the integers.
      2. Answers

    2. (Ratio-Geometry) The lengths of the sides of a triangle are in the ratio 2:3:4. The perimeter is 63 cm. Find the length of each side.
      3. Answers

    3. (Variation) Y varies jointly as x and the square of z, and inversely as the principal square root of w. If Y = 270 when x = 5, z = 6, and w = 4, what is Y when x = 3, z = 6, and w = 9?
      4. Answers

    4. (Motion) Steve traveled 340 miles. For most of the trip he drove 60 mph, but for one period of time he was slowed to 20 mph due to construction. If Steve's total travel time was 7 hours, how many miles did he drive at the reduced speed?
      5. Answers

    5. (Mixtures) The radiator in an automobile holds 14 quarts. How much pure anti-freeze should be mixed with a 20% anti-freeze solution to obtain a 40% mixture that will fill the tank?
      6. Answers

    6. (Investment, I=PRT) Greg has $17,000 invested in bonds that pay 13% annually. He also has money invested in a certificate of deposit at 17% annually. The total amount of interest he earns from both investments is the same as if all the money was invested in a single account at 14%. How much does he have invested in the certificate of deposit? Round your answer to the nearest dollar.
      7. Answers

    7. (Money-Coins) A collection of dimes and nickels is worth $3.75. If there are 55 coins in the collection, how many of each type of coin are there?
      8. Answers

    8. (Work) An inlet pipe can fill a barrel of wine in 6 hours, and an outlet pipe can empty it in 8 hours. Through an error both pipes are left on. How long will it take for the barrel to fill?
      9. Answers

    9. (Geometry) A rancher's pasture is rectangular and has an area of 84 square miles. The length is 5 miles more than the width. How much will it cost to enclose the pasture with a fence if fencing materials cost an average of #1.80 per linear foot? (5280 ft = 1 mile)
    Answers

  16. Identify each (a - p) as a line, circle, parabola, ellipse, hyperbola, or none of these.
    17. Answers

    a. 2x2 + 3y = 4i. 6x2 + 6y2 = 12
    b. 2x2 + 3y2 = 4j. 6x - 6y = 12
    c. 2x + 3y = 4k. 6x2 - 6y = 12
    d. 2x2 - 3y = 4l. 6x2 - 6y2 = 12
    e. 2x2 - 3y2 = 4m. 6x2 + 6y = -12
    f. 2x - 3y = 4n. 6x2 + 6y2 = -12
    g. 6x + 6y2 = 12o. 6x2 - 6y2 = -12
    h. 6x + 6y = 12p. 6x - 6y = -12

  17. Graph the following linear equations.
    18. Answers

    3x + y = 3c. 2x + 3 = 0
    2y - 3x = 0d. y = -2

  18. Write the equation of the line through the points (3, -4) and (-2, 2).
    19. Answers

  19. Find the midpoint of the line segment between the points (3, -4) and (1, 6).
    20. Answers

  20. Find the distance between the points (3, 2) and (-2, -10).
    21. Answers

  21. Find the slope of the line that goes through the points P(-5, 2) and Q(-1, -6).
    22. Answers

  22. Write the equation of a line parallel or perpendicular to a given line.
    23. Answers

    1. Write the equation of the line parallel to 2x + 5y = 10 and through (4, 7).

    2. Write the equation of the line perpendicular to 2x + 5y = 10 and through (4, 7).

  23. Find the vertex of a parabola from its equation below.
    24. Answers

    a) y = x2 + 2 b) y = x2 - 3x + 1 c) y = 2x2 - 8x + 10

  24. Find the center and radius of a circle from its equation.
    25. Answers

    1. x2 + y2 = 9

    2. x2 + 4x + y2 = 7

    3. x2 - 6x + y2 + 8y + 10 = 0

  25. Sketch the graph of a circle or a parabola.
    26. Answers

    Graph the conics in 23 (a), (c) and 24 (a), (c).

  26. Sketch the graph of an ellipse or hyperbola, centered at the origin.
    27. Answers

    Graph the ellipse or hyperbola, centered at (0, 0).

    a. 4x2 + 9y2 = 36c.
    b. d.

    The real solutions of a system of equations in 2 variables represents the points of intersection of the graphs of the equations.

     

  27. Solve the following systems. For (a) and (b) state the product of the coordinates, xy

    28. after you find the solution as an ordered pair (x,y).Answers

    a. b) c)

    An alternative way that question 27a (above) could be asked is the following:

    If x + 2y = 7 and x - y = - 2 then find xy

  28. Solve systems of non-linear equations:
    29. Answers

    a. b) c)

  29. Solve each of the following inequalities.
    30. Answers

    a. 2(x - 1) - 3(x + 2) > -4(5x - 6)d.
    b. e. | 2x+3 | < 7
    c. f.

  30. Graph the following system of inequalities.
    31. Answers

    4x - 5y _ 20 AND y + x > 0