Intermediate Algebra Questions
With Solutions and Explanations - sample 4

Full explanations of solutions for intermediate algebra questions in sample 4 are presented.

  1. (True or False)     x 2 and 2 x are like terms.

    Solution

    The statement "x 2 and 2 x are like terms" is FALSE because the two terms do not have the same power of x.

  2. (True or False)     x-3 and -3x are unlike terms.

    Solution

    The statement "x-3 and -3x are unlike terms" is TRUE because the two terms do not have the same power of x.

  3. (True or False)     1 / (x - 9) = 0 for x = 9.

    Solution

    Substitute x by 9 in the expression 1 / (x - 9).

    1 / (x - 9) = 1 / (9 - 9) = 1 / 0 = undefined

    The statement "1 / (x - 9) = 0 for x = 9" is FALSE.

  4. (True or False)     The set of ordered pairs {(0,0),(2,0),(3,0),(10,0)} represents a function.

    Solution

    All values of the x coordinates are different and therefore the set of ordered pairs represents a function. The statement "The set of ordered pairs {(0,0),(2,0),(3,0),(10,0)} represents a function" is TRUE.

  5. (True or False)     |a - b| = b - a if b - a < 0.

    Solution

    Recall that if x > 0, then

    | x | = x

    But if b - a < 0, then a - b > 0 and

    |a - b| = a - b

    The statement "|a - b| = b - a if b - a < 0" is FALSE.

  6. (True or False)     |x2 + 1| = x2 + 1.

    Solution

    Recall that if x > 0, then

    | x | = x

    Since x2 + 1 is positive for all values of real x, then

    | x2 + 1 | = x2 + 1

    The statement |x2 + 1| = x2 + 1" is TRUE.

  7. (True or False)     √(x - 5) 2 = x - 5.

    Solution

    For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x = - 4 and evaluate the right and left hand expressions.

    Left: √(x - 5) 2 = √(-4 - 5) 2

    = √81 = 9

    Right: x - 5 = - 4 - 5 = - 9

    The above statement is not true for x = - 4 and therefore the statement "√(x - 5) 2 = x - 5" is FALSE.

  8. (True or False)     (x - 2)(x + 2) = x2 - 4x - 4.

    Solution

    Expand (x - 2)(x + 2)

    (x - 2)(x + 2) = x2 + 2x - 2x - 4

    Group like terms

    (x - 2)(x + 2) = x2 - 4

    The statement "(x - 2)(x + 2) = x2 - 4x - 4" is FALSE.

  9. (True or False)     sqrt(x + 9) = sqrt(x) + sqrt(9), for all x real.

    Solution

    For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x = 16 and evaluate the right and left hand expressions.

    Left: sqrt(x + 9) = sqrt(16 + 9)

    = sqrt(25) = 5

    Right: sqrt(x) + sqrt(9) = sqrt(16) + sqrt(9)

    = 4 + 3 = 7

    The above statement is not true for x = 16 and therefore the statement "sqrt(x + 9) = sqrt(x) + sqrt(9), for all x real" is FALSE.

  10. (True or False)     |x - 3| = |x| + |3|, for all x real and negative.

    Solution

    Start with |x - 3|

    |x - 3| = |3 - x| = |3 + (- x)|

    The absolute value of the sum of two positive numbers is equal to the sum of the numbers. Since x is negative - x is positive and hence

    |3 + (- x)| = |3| + |-x| = |3| + |x|

    The statement "|x - 3| = |x| + |3|, for all x real and negative" is TRUE.


  11. (True or False)     (x + 2)3 = x3 + 23, for all x real.

  12. (True or False)     If k = 4, then the equation x2 - kx = -4 has one solution only.

  13. (True or False)     The discriminant of the equation: 2x2 - 4x + 9 = 0 is negative.

  14. (True or False)     The degree of the polynomial P(x) = (x - 2)(-x + 3)(x - 4) is equal to -3.

  15. (True or False)     The distance between the points (0 , 0) and (1 , 1) in a rectangular system of axes is equal to 1.

  16. (True or False)     The slope of the line 2x + 3y = -2 is negative.

  17. (True or False)     The relation 2y + x2 = 2 represents y as a function of x.

  18. (True or False)     The relation 2y + x2 = 2 represents x as a function of y.

  19. (True or False)     The relation |x| = |y| DOES NOT represent x as a function of y.

  20. (True or False)     The relation |x| = |y| DOES NOT represent y as a function of x.







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