Solutions and Explanations to Intermediate Algebra True/False Questions

Solutions and full explanations of intermediate algebra questions in Intermediate Algebra True/False Questions 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. Hence, \( |x + 1| \lt 0 \) is never true, and the statement 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 to the left of zero. Since \( |a| \lt |b| \), then \( a > b \).
   Subtract \( a \) from both sides: \( a - a > b - a \Rightarrow 0 > b - a \).
   Hence, the statement is TRUE.

3. (True or False)    The equation \( 2x + 7 = 2(x + 5) \) has one solution.

   Solution

   Solve: \( 2x + 7 = 2x + 10 \)
   Subtract \( 2x \): \( 7 = 10 \) (false).
   Therefore, no solution exists. The statement is FALSE.

4. (True or False)    The multiplicative inverse of \( -\frac{1}{4} \) is \( -\frac{1}{8} \).

   Solution

   The inverse of \( -\frac{1}{4} \) is \( \frac{1}{-\frac{1}{4}} = -4 \).
   So the statement is FALSE.

5. (True or False)    \( \frac{x}{2 + z} = \frac{x}{2} + \frac{x}{z} \)

   Solution

   Try \( x = 8 \), \( z = 2 \):
   Left side: \( \frac{8}{2 + 2} = 2 \)
   Right side: \( \frac{8}{2} + \frac{8}{2} = 4 + 4 = 8 \)
   Since \( 2 \ne 8 \), the statement is FALSE.

6. (True or False)    \( |-8| - |10| = -18 \)

   Solution

   Evaluate: \( |-8| - |10| = 8 - 10 = -2 \ne -18 \)
   The statement is FALSE.

7. (True or False)    \( \left(\frac{8}{4}\right) \div 2 = \frac{8}{\left(4 \div 2\right)} \)

   Solution

   Left: \( 2 \div 2 = 1 \)
   Right: \( 8 \div 2 = 4 \)
   Since \( 1 \ne 4 \), the statement is FALSE.

8. (True or False)    \( 31.5(1.004)^{20} \lt 31.6(1.003)^{25} \)

   Solution

   Use calculator:
   Left: \( 31.5(1.004)^{20} \approx 34.118 \)
   Right: \( 31.6(1.003)^{25} \approx 34.057 \)
   Since \( 34.118 > 34.057 \), the statement is FALSE.

9. (True or False)    The graph of the equation \( y = 4 \) has no x-intercept.

   Solution

   The line \( y = 4 \) is horizontal and does not cross the x-axis.
   Hence, the statement is TRUE.

10. (True or False)    \( \frac{n(n+3)}{2} = \frac{3}{2} \) when \( n = 0 \)

    Solution

    Evaluate: \( \frac{0(0 + 3)}{2} = 0 \)
    Since \( 0 \ne \frac{3}{2} \), the statement is FALSE.

11. (True or False)    The distance between -9 and 20 is equal to the distance between 9 and -20.

    Solution

    Distance formula: \( |a - b| = |b - a| \)
    \( |-9 - 20| = |-29| = 29 \)
    \( |9 - (-20)| = |9 + 20| = 29 \)
    So the distances are equal. The statement is TRUE.

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

    Solution

    Evaluate: \( f(-3) = \sqrt{1 - (-3)} = \sqrt{4} = 2 \)
    So the statement is TRUE.

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

    Solution

    Rewriting in slope-intercept form:
    \( 2x + 2y = 2 \Rightarrow 2y = -2x + 2 \Rightarrow y = -x + 1 \)
    Slope = \( -1 \), so the statement is FALSE.

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

    Solution

    The absolute value of a number is always non-negative, meaning it is either positive or zero.
    For example, when \( x = -5 \), we have \( |x + 5| = |0| = 0 \).
    Since \( |x + 5| \) can be zero, it is not *always* positive.
    Hence, the statement is FALSE.

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

    Solution

    Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
    \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
    Substitute: \( \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25} = 5 \)
    Hence, the statement is TRUE.

16. (True or False)    \( \frac{1}{2x - 4} \) is undefined when \( x = -4 \).

    Solution

    Substitute \( x = -4 \):
    \[ \frac{1}{2(-4) - 4} = \frac{1}{-8 - 4} = \frac{1}{-12} \]
    This is defined. But the expression is **undefined** when the denominator is zero:
    \[ 2x - 4 = 0 \Rightarrow x = 2 \]
    So at \( x = -4 \), it is defined.
    Hence, the statement is FALSE.

17. (True or False)    \( \left(-\frac{1}{5}\right)^{-2} = 25 \)

    Solution

    A negative exponent means reciprocal:
    \[ \left(-\frac{1}{5}\right)^{-2} = \left(-5\right)^2 = 25 \]
    Hence, the statement is TRUE.

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

    Solution

    The reciprocal of a number \( x \) is \( \frac{1}{x} \).
    But \( \frac{1}{0} \) is undefined in mathematics.
    So the reciprocal of 0 is not defined, let alone equal to 0.
    Hence, the statement is FALSE.

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

    Solution

    The additive inverse of a number \( a \) is the number that when added to \( a \) gives zero.
    So the additive inverse of \( -10 \) is \( 10 \), since \( -10 + 10 = 0 \).
    Hence, the statement is TRUE.

20. (True or False)    \( \frac{1}{x - 4} = \frac{1}{x} - \frac{1}{4} \)

    Solution

    Let us test with \( x = 8 \):
    Left side: \( \frac{1}{8 - 4} = \frac{1}{4} \)
    Right side: \( \frac{1}{8} - \frac{1}{4} = \frac{1 - 2}{8} = -\frac{1}{8} \)
    Since \( \frac{1}{4} \ne -\frac{1}{8} \), the statement is FALSE.