Solutions and Explanations
to Intermediate Algebra Questions in Sample 1

Detailed solutions and full explanations of intermediate algebra questions in sample 6 are presented.

1. (True or False)   ((2²)³) = 2⁵.
   Solution
   Evaluate ((2²)³) and 2⁵ .
   ((2²)³) = 2⁶ = 64
   2⁵ = 32
   The statement "((2²)³) = 2⁵." is FALSE.
   

2. (True or False)   The slope of a vertical line is undefined.
   Solution
   Let (2 , 5) and (2 , 7) being two points on the same vertical line since their x coordinates are both equal to 2. Let us find the slope through the two points.
   m = (7 - 5) / (2 - 2) = 2 / 0 = undefined.
   The statement "the slope of a vertical line is undefined" is TRUE.
   

3. (True or False)   Two lines with positive slopes can be perpendicular.
   Solution
   For two lines to be perpendicular, the product of their slopes must be equal to - 1. If the two slopes are positive their product cannot be negative. Hence the statement "Two lines with positive slopes can be perpendicular" is FALSE.
   

4. (True or False)   5 > 10 or 5 < 7.
   Solution
   The statement "5 > 10" is false and the statement "5 < 7" is true. The whole statement "5 > 10 or 5 < 7" is TRUE since there is an "or" and one of the two statements is true.
   

5. (True or False)   The set of ordered pairs {(6,4),(2,-5),(-2,4),(6,-4)} is a function.
   Solution
   For the first coordinate 6 there are two possible second coordinates: 4 and - 4 and therefore the statement "the set of ordered pairs {(6,4),(2,-5),(-2,4),(6,-4)} is a function" is FALSE.
   

6. (True or False)   The additive inverse of -10 is 10.
   Solution
   The additive inverse of a number x is.
   - x
   The additive inverse of - 10 is.
   - (-10) = 10
   The statement "the additive inverse of -10 is 10" is TRUE.
   

7. (True or False)   The product of two positive numbers is NOT positive.
   Solution
   The product of two positive numbers is positive and The statement "the product of two positive numbers is NOT positive" is FALSE.
   

8. (True or False)   x + 2 = 7 is called an inequality.
   Solution
   x + 2 = 7 has an equal sign and is called an equation. Hence the statement "x + 2 = 7 is called an inequality" is FALSE.
   

9. (True or False)   The associative property is used to write 4x + 8y = 4(x + 2y).
   Solution
   No. The distributive property is used to write
   4x + 8y = 4(x + 2y)
   The statement "the associative property is used to write 4x + 8y = 4(x + 2y)" is FALSE
   

10. (True or False)   The absolute value of a real negative number is negative.
    Solution
    The absolute value of a real negative number is positive
    

11. (True or False)   -2³ = (-2)³.
    Solution
    Evaluate each expression
    -2³ = - 2 * 2 * 2 = - 8
    (-2)³ = (-2)(-2)(-2) = - 8
    

12. (True or False)   30% of x is equal to 0.03x
    Solution
    30% of x is written as
    30% × x = (30/100) x = 0.03 x
    

13. (True or False)   "x is at most equal to 9" is written mathematically as x < 9.
    Solution
    No. It is written as
    x ≤ 9
    

14. (True or False)   3²⁰ + 3²⁰ + 3²⁰ = 3²¹
    Solution
    factor and simplify
    3²⁰ + 3²⁰ + 3²⁰ = 3×3²⁰ = 3²¹
    

15. (True or False)   1.5 × 10^-5 is the scientific notation of the number 0.0000015.
    Solution
    Convert 1.5 × 10^-5 into decimal
    1.5 × 10^-5 = 1.5 / 10⁵ = 1.5 / 100000 = 0.000015
    

16. (True or False)   1000/0 = 0.
    Solution
    Undefined; division by zero not allowed in mathematics.
    

17. (True or False)   0/1000 = 0.
    Solution
    Zero divided by any non zero number gives zero
    

18. (True or False)   0.0000001⁰ = 1.
    Solution
    Any non zero number to the power zero gives 1.
    

19. (True or False)   1 / (( -2 ) ^-4) = 16.
    Solution
    Rewrite 1 / (( -2 ) ^-4) with positive power.
    1 / (( -2 ) ^-4) = (-2) ⁴ = (-2)(-2)(-2)(-2) = 16
    

20. (True or False)   x = 7 DOES NOT satisfy the inequality x - 7 < 0
    Solution
    Substitute x by 7 in the inequality.
    7 - 7 < 0 which gives 0 < 0
    The above statement is false and therefore x = 7 does not satisfy the inequality.

1. FALSE (2⁶)
2. TRUE
3. FALSE (m1*m2 = - 1 < 0)
4. TRUE
5. FALSE (note the pairs (6,4) and (6,-4))
   
6. TRUE
7. FALSE (positive * positive = positive)
8. FALSE (equation)
9. FALSE (distributive property)
10. FALSE (positive or zero)
    
11. TRUE
12. FALSE (0.3x)
13. FALSE (x ≤ 9)
14. TRUE
15. FALSE (0.000015)
    
16. FALSE (division by 0 is undefined in mathematics)
17. TRUE
18. TRUE
19. TRUE
20. TRUE