Answers to Math 390 Semester Review Sheet



1.  Find the value of each expression>

    a.  [(4)(-8) + (-6)² ]/[(-2)(-2)] = 1

    b.  5(4 + 3) + 6[14 - 6(2 - 3)] = 155

2.  Add or subtract.

    a.  (6x³ + 3x² + 2) - (5x³  + 4x² + 3) - (5x² - 4x - 6)

         = x³  - 6x²  + 4x + 5

    b.  4/(8xy² ) - 2/(12x² y) = (3x - y)/(6x² y² )


    c.  5/(x² - 16) - 2/(x² - 3x - 4) + 3/(x² - 1)

        = (6x²  - 6x - 45)/[(x + 4)(x - 4)(x + 1)(x - 1)]


    d.  (4x²  - 3) + (3x²  - 4x - 4) = 7x²  - 4x - 7

    e.  (x + 5)/(x - 10) + (3x - 2)/(x - 10) = (4x + 3)/(x - 10)

3.  Multiply or divide.

    a.  8x² y³ (2x²  + 3y² )

        = 16x to the fourth y cubed + 24 x squared y to the fifth

    b.  (2xy³ )(x² y)(7xy) = 14 x to the fourth y to the fifth

    c.  (x² y - z)(x³ y²  - x² yz)

        = x to the fifth y cubed -  x to the fourth y squared z - x cubed y square z + x² yz²


    d.  (2x + 1)(3x - 2) = 6x²  - x - 2

    e.  (ab - 3)(ab + 3) = a² b²  - 9

    f.  (x + 5)² (x - 1) = x³  + 9x²  + 15x - 25

    g.  [(x - 3)/(2x + 1)][(4x² - 1)/(x² - 5x + 6)]

        = (2x - 1)/(x - 2)

    h.  (36a³ b³ c²)/(6ab³ c³ ) = 6a² /c

    i.  (x - y)/(y - x) = -1

    j.  [(x²  - 25)/(x²  + 7x + 12)][(x - 2)/(4x²  - 9)]divided by [(x - 3)/(2x²  + 11x + 12)]

        Remark:  I do not know how to print the symbol used in arithmetic to denote division.

        Answer: [(x + 5)(x - 5)(x - 2)]/[(x + 3)(2x - 3)(x - 3)]

    k.  (64x³ y² - 32xy³ )/(8xy² ) = 8x² - 4y

    l.  (3x²  + 6x + 1)/(x - 2) = 3x + 12 + 25/(x - 2)

4.  Factor completely.

    a.  2m² n³  - 6m² n²  + 4m³ n²  = 2m² n² (n - 3 + 2m)

    b.  x²  - 2x - 8 = (x - 4)(x + 2)

    c.  2x³ y + 8x² y - 24xy = 2xy(x + 6)(x + 2)

        Remember to look for a common factor first.

    d.  x²  + 14x + 49 = (x + 7)²


    e.  4y²  - 169 = (2y + 13)(2y - 13)

    f.  21x²  + 4x - 1 = (7x - 1)(3x + 1)

    g.  9x² - 12x + 4 = (3x - 2)²


    h.  8x³  + 125 = (2x + 5)(4x²  - 10x + 25)

    i.  27a³ b³  - 64 = (3ab - 4)(9a² b²  + 12ab + 16)

5.  Solve each of the following equations or inequalities.

    a.  y + 9 = 18  Answer: y = 9

    b.  17x + 8(1 - 2x) = 2/3  Answer:  x = -22/3

    c.  6(x + 2) = 4 + 5(x - 4)  Answer:  x = -28

    d.  8y - 18 = 2y + 30  Answer:  y = 8

    e.  3 > 4 - 2x  Answer:  x >  1/2

    f.  (x - 1)/x = (x + 1)/(x - 1)  Answer: x = 1/3

    g.  1/4 - 2/x = 3/2 Answer x = -8/5

    h.  -4 <  5 - 3x  Answer x <  3

    i.  7/t = 14  Answer t = 1/2

6.  Solve the word problems.  The problems are so easy that they
    can be solved using arithmetic.  Since we are studying algebra
    I will expect an algebraic solution.

    a.  If two more than a number is equal to four less than three
        times the number, find the number.

        The number is three.

    b.  If a rug is 3 feet longer than it is wide and its perimeter
        is 38 feet, find its dimensions.

        The length is 11 feet, the width 8 feet.

    c.  If two consecutive odd integers sum to 24, find the integers.

        The integers are 11 and 13.

    d.  A triangle whose base is 16 cm has an area of 144 square
        centimeters.  Find the altitude of the triangle.

        The altitude is 18 cm.
        
    e.  If you drive for 2 hours and go 110 miles, what is your 
        average speed?

        Average speed is 55 mph.

    f.  How many quarts of a 40% salt solution must be added to 40
        quarts of a 10% salt solution to obtain a 20% salt solution?

        The answer is 20 quarts.

    g.  An amount of money is invested at 9% and $1200 more than that
        amount is invested at 10%.  How much income is received from
        each investment if the total income from the two investments 
        is $1013?

        Income on 9% investment is $423 and on 10% investment is $590.

    i.  A woman drives 120 miles in the same time a man drives 80 miles.
        If the speed of the woman is 20 miles per hour greater than the
        speed of the man, find the speed of each.

        Speed of woman is 60 mph.  The speed of the man is 40 mph.

7.  Find the equation of the line that passes through (5,2) and (-3,-2).
    Express answer in slope-intercept form.

    Answer: y = x/2 - 1/2

8.  Find the equation of the line that passes through (6,2) which is
    parallel to the line y = (1/3)x + 5.  Express answer in standard
    form, that is of the form Ax + By + c = 0 where A, B, and C are
    integers.

    Abswer:  x - 3y = 0

9.  Find the equation of the line that passes through (-1,-5) and is 
    perpendicular to the line x + 2y = 2.  Express your answer in standard
    form, that is of the form Ax + By + c = 0 where A, B, and C are
    integers.

    Answer: 2x - y - 3 = 0

10.  Use the intercept method to graph the line 2x + y = 2.

     The x-intercept is 1 and the y-intercept is 2.

11.  Graph the line 6x - 3y = 12.

     The x-intercept is 2 and the y-intercept is -4.

12.  Graph the inequality x - 3y > -6.

     Graph consists of all points on or below the line x - 3y = -6.

13.  Write in scientific notation.

     a.  0.05026 = 5.026(10-² )

     b.  99.103 = 9.9103(10)

14.  Simplify the following radical expressions.
     Notation:  Sqrt (x) denotes the principal square root of x.
     In what follows all variables are positive.

     a.  Sqrt (36/9x² ) = 6/(3x² )

     b.  Sqrt (98xy³ z² ) = 7yz times Sqrt(2xy)

     c.  Sqrt (49x³ ) = 7x times Sqrt(x)

     d.  2(Sqrt(6)) - 2(Sqrt(24)) + Sqrt(54) = Sqrt(6)

     e.  Sqrt(18)/2 - 2(Sqrt(2))/3 = [5 times Sqrt(2)]/6

     f.  Sqrt(2)(Sqrt(6) + 3) = 2 times Sqrt(3) + 3 times Sqrt(2)

     g.  (Sqrt(3) + 2)(2(Sqrt(3)) - 1) = 4 + 3 times Sqrt(3)

     h.  Sqrt((8x)/(3y)) = [2 times Sqrt(6xy)]/[3y]

     i.  (5x)/Sqrt(3) = [5x times Sqrt(3)]/3

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