Intermediate Algebra Questions With Solutions and Explanations - sample 4

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

(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.

(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.

(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.

(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.

(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.

(True or False) |x^{2} + 1| = x^{2} + 1.

Solution

Recall that if x > 0, then

| x | = x

Since x^{2} + 1 is positive for all values of real x, then

| x^{2} + 1 | = x^{2} + 1

The statement |x^{2} + 1| = x^{2} + 1" is TRUE.

(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.

(True or False) (x - 2)(x + 2) = x^{2} - 4x - 4.

Solution

Expand (x - 2)(x + 2)

(x - 2)(x + 2) = x^{2} + 2x - 2x - 4

Group like terms

(x - 2)(x + 2) = x^{2} - 4

The statement "(x - 2)(x + 2) = x^{2} - 4x - 4" is FALSE.

(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.

(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.

(True or False) (x + 2)^{3} = x^{3} + 2^{3}, for all x real.

(True or False) If k = 4, then the equation x^{2} - kx = -4 has one solution only.

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

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

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

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

(True or False) The relation 2y + x^{2} = 2 represents y as a function of x.

(True or False) The relation 2y + x^{2} = 2 represents x as a function of y.

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

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