Detailed Solutions of True or False Questions Intermediate Algebra

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

1. (True or False)   \( ((2^2)^3) = 2^5 \)

   Solution

   Evaluate \( ((2^2)^3) \) and \( 2^5 \).
   \( ((2^2)^3) = 2^6 = 64 \)
   \( 2^5 = 32 \)
   The statement "\( ((2^2)^3) = 2^5 \)" is FALSE.
   

2. (True or False)   The slope of a vertical line is undefined.

   Solution

   Let (2 , 5) and (2 , 7) be two points on the same vertical line. Let us find the slope.
   \( m = \frac{7 - 5}{2 - 2} = \frac{2}{0} \) = undefined.
   The statement is TRUE.
   

3. (True or False)   Two lines with positive slopes can be perpendicular.

   Solution

   If both slopes are positive, their product cannot be \(-1\). Hence the statement is FALSE.
   

4. (True or False)   \( 5 > 10 \) or \( 5 < 7 \)

   Solution

   "5 > 10" is false and "5 < 7" is true. Since "or" is used, the whole statement is TRUE.
   

5. (True or False)   The set of ordered pairs \(\{(6,4),(2,-5),(-2,4),(6,-4)\}\) is a function.

   Solution

   The x-value 6 maps to two y-values. Hence the statement is FALSE.
   

6. (True or False)   The additive inverse of \(-10\) is 10.

   Solution

   The additive inverse of \(x\) is \(-x\).
   So the additive inverse of \(-10\) is \(-(-10) = 10\)
   The statement is TRUE.
   

7. (True or False)   The product of two positive numbers is NOT positive.

   Solution

   Product of two positive numbers is positive. The statement is FALSE.
   

8. (True or False)   \(x + 2 = 7\) is called an inequality.

   Solution

   An equal sign makes it an equation. The statement is FALSE.
   

9. (True or False)   The associative property is used to write \(4x + 8y = 4(x + 2y)\).

   Solution

   This uses the distributive property, not associative. Statement is FALSE.
   

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

    Solution

    It is positive. The statement is FALSE.
    

11. (True or False)   \( -2^3 = (-2)^3 \)

    Solution

    Evaluate both sides:
    \( -2^3 = -(2^3) = -8 \)
    \( (-2)^3 = -8 \)
    

12. (True or False)   30% of \(x\) is equal to \(0.03x\)

    Solution

    30% of \(x\) is \( \frac{30}{100}x = 0.3x \). Statement is FALSE.
    

13. (True or False)   "x is at most equal to 9" is written mathematically as \(x < 9\).

    Solution

    It should be \( x \leq 9 \). Statement is FALSE.
    

14. (True or False)   \(3^{20} + 3^{20} + 3^{20} = 3^{21}\)

    Solution

    Factor and simplify:
    \( 3^{20} + 3^{20} + 3^{20} = 3 \cdot 3^{20} = 3^{21} \). Statement is TRUE.
    

15. (True or False)   \(1.5 \times 10^{-5}\) is the scientific notation of 0.0000015

    Solution

    \(1.5 \times 10^{-5} = 0.000015\). Statement is FALSE.
    

16. (True or False)   \( \frac{1000}{0} = 0 \)

    Solution

    Undefined; division by zero is not allowed. Statement is FALSE.
    

17. (True or False)   \( \frac{0}{1000} = 0 \)

    Solution

    True. Zero divided by any nonzero number is zero.
    

18. (True or False)   \( 0.0000001^0 = 1 \)

    Solution

    Any non-zero number to the zero power is 1. Statement is TRUE.
    

19. (True or False)   \( \frac{1}{(-2)^{-4}} = 16 \)

    Solution

    Rewrite with positive exponent:
    \( \frac{1}{(-2)^{-4}} = (-2)^4 = 16 \). Statement is TRUE.
    

20. (True or False)   \(x = 7\) DOES NOT satisfy the inequality \(x - 7 \lt 0\)

    Solution

    Substitute \(x = 7\): \(7 - 7 \lt 0\) gives \(0 \lt 0\), which is false.
    So the original statement is TRUE.