Solutions to Algebra Placement Test Practice

Solutions with explanations to algebra placement test practice.

  1. Solution
    Given expression
    (a3 - 1) / (b - 1)
    Substitute a by -2 and b by -2 in given expression
    ((-2)3 - 1) / ((-2) - 1)
    Simplify expression above
    (-8 - 1) / (-2 - 1) = 3



  2. Solution
    Given equation
    - 2 (x + 9) = 20
    Solve for x
    - 2 x - 18 = 20
    - 2 x = 38
    Multiply both sides of the above equation
    2( -2 x) = 2( 38)
    Simplify to obtain
    - 4 x = 76



  3. Solution
    x is a variable
    20% of x is written as
    20% x
    the fifth of x is written as
    (1 / 5) x
    "20% of it is added to its fifth" is written as:
    20% x + (1 / 5) x
    seven tenths of x is written as
    (7 / 10) x = 0.7 x
    the result is equal to 12 subtracted from seven tenths of x
    20% x + (1 / 5) x = 0.7x - 12
    Change % and fractions to decimal numbers
    0.2 x + 0.2 x = 0.7x - 12
    Solve for x
    12 = 0.3 x
    x = 40



  4. Solution
    Given formula
    A = 0.5 (b + B) h
    Divide both sides of the formula by 0.5 h
    (A / (0.5 h)) = 0.5 (b + B) h / (0.5 h)
    simplify
    (A / (0.5 h)) = b + B
    Solve for B
    B = (A / 0.5 h) - b
    Note that 1 / 0.5 = 2, hence
    B = 2 A / h - b



  5. Solution
    Let x be the amount of money in dollars that Tom has
    Alex has the third of Tom
    (1 / 3) x
    Linda has twice as much as Alex
    2 (1 / 3) x
    All together they have 120 dollars, hence
    x + x / 3 + 2 x / 3 = 120
    Multiply all terms of the above equation by 3, simplify and solve for x
    3 x + x + 2 x = 360
    x = 60
    Linda has
    2 (1 / 3) x = (2 / 3) 60 = $40



  6. Solution
    Given
    6 x2 - 11 x - 2
    The linear terms that will make the factors has coefficients 6 and 1 (since 6*1 = 6) or 2 and 3 (since 2*3=6)
    (6x     )(x      )

    (2x      )(3x     )
    The constant terms should -1 and 2 or 1 and -2
    (6 x - 1)(x + 2)
    If (6x - 1)(x + 2) is expanded, it does not give 6 x2 - 11 x - 2.
    (6 x - 1)(x + 2) = 6 x2 + 11 x - 2
    We now try (6x + 1)(x - 2) which when expanded gives 6x2 - 11 x - 2
    (6 x + 1)(x - 2) = 6 x2 - 11 x - 2



  7. Solution
    Given
    x2 - 7 x - 8
    Factor
    (x + 1)(x - 8)
    One of the listed factors is
    x + 1



  8. Solution
    Given
    (2xy2 - 3x2y) - (2x2y2 - 4x2y)
    Eliminate brackets taking signs into consideration
    = 2xy2 - 3x2y - 2x2y2 + 4x2y
    Group like terms - 3x2y and 4x2y
    = 2xy2  - 2x2y2 + x2y



  9. Solution
    Average speed is given by
    Total distance / total time = 70 miles / hour
    Total distance is given by
    x + 200 miles
    Total time is equal to
    2 + 3 = 5 hours
    Substitute in formula above
    (x + 200) / 5 = 70
    Solve above equation for x
    x + 200 = 350
    x = 150 miles



  10. Solution
    Rewrite all 4 equations in slope intercept form
    (I) y = (- 3 / 2) x + 3 / 2
    (II) y = (-2 / 3) x - 5 / 3
    (III) y = (2 / 3) x - 3 / 2
    (IV) y = - 3x + 9 / 2
    The slopes of all 4 lines are
    (I) -3/2
    (II) -2/3
    (III) 2/3
    (IV) -3
    For two lines with slopes m and n to be perpendicular, the product of their slopes must equal -1 or m*n = -1.
    The product of the slopes of equations (I) and (III) is given by

    (-3 / 2)*(2 / 3) = - 1
    hence (I) and (III) are perpendicular.



  11. Solution
    Given
    f (x) = (x + 1)2
    To find f (t + 2), we substitute x by t + 2 in f (x); hence
    f (t + 2) = ((t + 2)+ 1) 2
    then simplify and expand
    = (t + 3) 2
    = t 2 + 6t + 9



  12. (√x + √y) (√x - √y) - (√x - √y)2 =


    Solution
    Given
    (√x + √y) (√x - √y) - (√x - √y) 2
    Expand (√x + √y) (√x - √y)
    (√x + √y) (√x - √y) = (√x) 2 - (√y) 2 = x - y
    Expand (√x - √y)2
    (√x - √y)2 = (√x) 2 + (√y) 2 - 2√x √y = x + y - 2√x √y
    Hence
    (√x + √y) (√x - √y) - (√x - √y) 2
    = (x - y) - (x + y - 2√x √y)
    = - 2 y + 2√x √y



  13. x / 2 - y / 4 = 7

    Solution
    Equation of line given in standard form
    x / 2 - y / 4 = 7
    To determine the slope, we need to write the given equation in slope intercept form. Multiply all terms by 4 and simplify and solve for y.
    2 x - y = 28
    y = 2x - 28
    Slope is the coefficient of x and is equal to
    2



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

    Solution
    Given
    (x / (x - 3) + 1 / 2) (2 / (x - 1))
    We first simplify the expression ( x / (x - 3) + 1 / 2). Reduce to the lowest common denominator.
    ( x / (x - 3) + 1 / 2)
    2x / ( 2 (x - 3) ) + ( x- 3)  / (2 (x - 3))
    = (3x - 3) / (2(x - 3))
    Hence
    x / (x - 3) + 1 / 2) (2 / (x - 1))

    = (3x - 3) (2/(x - 1))) / (2(x - 3))
    = 3(x - 1)(2 / (x - 1)) / (2(x- 3))
    Simplify
    = 3 / (x - 3)



  15. Solution
    Let a and b be the x and y coordinates of point B, hence
    M is the midpoint of A and B
    Since the midpoint M(3,2) is known and point A(2,1) is also known, we can write
    (3,2) = ( (2 + a)/2 , (1 + b)/2 )
    Two equations may be written as follows
    3 = (2 + a) / 2 and 2 = (1 + b) / 2
    Solve for a and b
    a = 4 and b = 3 |>

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