Solutions and Explanations to Intermediate Algebra Questions in Sample 2

Solutions and full explanations of intermediate algebra questions in sample 2 are presented.

  1. (True or False)     The inequality |x + 1| < 0 has no solution.
    Solution
    The absolute value of a real expression is either positive or equal to zero. Therefore there is no value of x that makes |x + 1| negative and therefore |x + 1| < 0 is never true and the statement "The inequality |x + 1| < 0 has no solution" is TRUE.

  2. (True or False)     If a and b are negative numbers, and |a| < |b|, then b - a is negative.
    Solution
    Since a and b are both negative, they are positioned to the left of zero on the number line. Since |a| < |b|, a is closer to zero than b and therefore a is greater than b which written as
    a > b
    Subtract a to both sides and simplify
    a - a > b - a
    0 > b - a
    Hence the statement "b - a is negative" is TRUE.

  3. (True or False)     The equation 2x + 7 = 2(x + 5) has one solution.
    Solution
    Let us solve the given equation
    2x + 7 = 2(x + 5)
    2x + 7 = 2x + 10 , expand right hand term
    2x + 7 - 2x = 2x + 10 - 2x , subtract 2x from both sides
    7 = 10 , simplify
    The above statement is never true and therefore the given equation has no solutions. The statement "The equation 2x + 7 = 2(x + 5) has one solution" is FALSE.

  4. (True or False)     The multiplicative inverse of -1/4 is -1/8.
    Solution
    If a real number x is not equal to zero, its multiplicative inverse is equal to 1/x. Hence the inverse of -1/4 is equal to
    1 / (-1/4) = (1/1) / (-1/4) = (1/1)*(-4/1) = - 4
    and therefore the statement "The multiplicative inverse of -1/4 is -1/8" is FALSE.

  5. (True or False)     x (2 + z) = x 2 + x z
    Solution
    let us use the values x = 8 and z = 2 and evaluate the values of the left side and right side expressions.
    Left side: x (2 + z) = 8 (2 + 2) = 2
    Right side: x 2 + x z = 8 2 + 8 2 = 4 + 4 = 8
    Since x (2 + z) = x 2 + x z is not true for one value of x and one value of z, the statement is FALSE.

  6. (True or False)     |-8| - |10| = -18
    Solution
    Evaluate left side.
    |-8| - |10| = 8 - 10 = - 2
    hence the statement |-8| - |10| = -18 is FALSE.

  7. (True or False)     (8 4) 2 = 8 (4 2)
    Solution
    Evaluate left side.
    (8 4) 2 = 2 2 = 1
    Evaluate right side.
    8 (4 2) = 8 2 = 4
    hence the statement (8 4) 2 = 8 (4 2) is FALSE.

  8. (True or False)     31.5(1.004)20 < 31.6(1.003)25
    Solution
    Use calculator and calculate left and right sides of inequality.
    31.5(1.004)20 = 34.118 (rounded to 3 decimal places)
    31.6(1.003)25 = 34.057 (rounded to 3 decimal places)
    since 34.118 is greater than 34.057 the statement 31.5(1.004)20 < 31.6(1.003)25 is FALSE.

  9. (True or False)     The graph of the equation y = 4 has no x-intercept.
    Solution
    The line with equation y = 4 is a horizontal line parallel to the x axis and hence cannot intersect the x axis. The statement " the graph of the equation y = 4 has no x-intercept" is TRUE.

  10. (True or False)     The value of n(n + 3)/2 = 3/2 when n = 0.
    Solution
    Evaluate n(n + 3)/2 = 3/2 for n = 0.
    n(n + 3)/2 = 0(0 + 3) / 2 = 0 / 2 = 0
    The statement "the value of n(n + 3)/2 = 3/2 when n = 0" is FALSE.

  11. (True or False)     The distance between the numbers -9 and 20 is equal to the distance between 9 and -20 on the number line.
    Solution
    Distance between numbers a and b on a number line is given by .
    |a - b| = |b - a|
    Hence, distance between -9 and 20 is equal to
    |-9 - 20| = 29
    Hence, distance between 9 and - 20 is equal to
    |9 - (-20)| = 29
    The two distances are equal.

  12. (True or False)     If f(x) = √(1 - x), then f(-3) = 2.
    Solution
    f(-3) = √(1 - (-3)) = √4 = 2

  13. (True or False)     The slope of the line 2x + 2y = 2 is equal to 2.
    Solution
    Write equation in slope intercept y = m x + b form and identify its slope m. Given equation
    2x + 2y = 2
    Subtract 2 x from both sides and simplify
    2 y = - 2x + 20
    Divide all terms by 2
    y = - x + 10
    Slope is equal to
    - 1

  14. (True or False)     |x + 5| is always positive.

  15. (True or False)     The distance between the points (0 , 0) and (5 , 0) in a rectangular system of axes is 5.

  16. (True or False)     1 / (2x - 4) is undefined when x = -4.

  17. (True or False)     (-1/5)-2 = 25.

  18. (True or False)     The reciprocal of 0 is equal to 0.

  19. (True or False)     The additive inverse of -10 is equal to 10.

  20. (True or False)     1 / (x - 4) = 1/x - 1/4.

Answers to the Above Questions
  1. TRUE
  2. TRUE
  3. FALSE (2x + 7 = 2x + 10 , 7 = 10 no solution)
  4. FALSE ( multiplicative inverse of -1/4 is -4)
  5. FALSE (try the values x = 8 and z = 2)
  6. FALSE ( = 8 - 10 = -2)
  7. FALSE (left side = 1 , right side = 4)
  8. FALSE
  9. TRUE
  10. FALSE ( =0 )
  11. TRUE
  12. TRUE
  13. FALSE ( = -1 )
  14. FALSE ( = 0 for x = -5)
  15. TRUE
  16. FALSE (undefined when 2x - 4 = 0 , x = 2)
  17. TRUE
  18. FALSE (the reciprocal of 0 is undefined)
  19. TRUE
  20. FALSE (try x = 2)

Algebra Questions and problems
More ACT, SAT and Compass practice