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 xintercept.
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 xintercept" 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.

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

