# 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| %lt; 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| %lt; 0 is never true and the statement "The inequality |x + 1| %lt; 0 has no solution" is TRUE.

2. (True or False)     If a and b are negative numbers, and |a| %lt; |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| %lt; |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)

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