Intermediate Algebra Questions With Solutions and Explanations - sample 2

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

(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.

(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.

(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.

(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.

(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.

(True or False) |-8| - |10| = -18

Solution

Evaluate left side.

|-8| - |10| = 8 - 10 = - 2

hence the statement |-8| - |10| = -18 is FALSE.

(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.

(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.

(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.

(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.

(True or False) The distance between the numbers -9 and 20 is equal to the distance between 9 and -20 on the number line.

(True or False) If f(x) = sqrt(1 - x), then f(-3) = 2.

(True or False) The slope of the line 2x + 2y = 2 is equal to 2.

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

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

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

(True or False) (-1/5)^{-2} = 25.

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

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