Intermediate Algebra True/False Questions with Solutions

Problem: (True or False)    The inequality \( |x + 1| < 0 \) has no solution.

Solution:

The absolute value of a real expression is always positive or equal to zero (\( \ge 0 \)). Therefore, there is no value of \( x \) that makes \( |x + 1| \) negative. Hence, \( |x + 1| < 0 \) is never true.
Answer: TRUE

Problem: (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 numbers, their absolute values are given by their opposites: \( |a| = -a \) and \( |b| = -b \).
Thus, the condition \( |a| < |b| \) can be rewritten analytically as: \[ -a < -b \] Multiply both sides by \( -1 \) (which reverses the inequality direction): \[ a > b \] Now, subtract \( a \) from both sides to isolate the expression \( b - a \): \[ 0 > b - a \quad \text{or} \quad b - a < 0 \] Since \( b - a \) is less than zero, it is negative.
Answer: TRUE

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

Solution:

Expand and simplify the equation: \[ 2x + 7 = 2x + 10 \] Subtract \( 2x \) from both sides: \[ 7 = 10 \] This is a contradiction, meaning no solution exists.
Answer: FALSE

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

Solution:

The multiplicative inverse of a number \( a \) is \( \dfrac{1}{a} \). For \( -\dfrac{1}{4} \): \[ \dfrac{1}{-\dfrac{1}{4}} = -4 \] Answer: FALSE

Problem: (True or False)    \( \dfrac{x}{2 + z} = \dfrac{x}{2} + \dfrac{x}{z} \)

Solution:

Test with counterexample values \( x = 8 \) and \( z = 2 \):
Left side: \( \dfrac{8}{2 + 2} = \dfrac{8}{4} = 2 \)
Right side: \( \dfrac{8}{2} + \dfrac{8}{2} = 4 + 4 = 8 \)
Since \( 2 \neq 8 \), the statement is incorrect.
Answer: FALSE

Problem: (True or False)    \( |-8| - |10| = -18 \)

Solution:

Evaluate the absolute values: \[ |-8| - |10| = 8 - 10 = -2 \neq -18 \] Answer: FALSE

Problem: (True or False)    \( \left(\dfrac{8}{4}\right) \div 2 = \dfrac{8}{\left(4 \div 2\right)} \)

Solution:

Evaluate each side separately:
Left side: \( 2 \div 2 = 1 \)
Right side: \( 8 \div 2 = 4 \)
Since \( 1 \neq 4 \), division is not associative.
Answer: FALSE

Problem: (True or False)    \( 31.5(1.004)^{20} < 31.6(1.003)^{25} \)

Solution:

Evaluating both sides:
Left side: \( 31.5(1.004)^{20} \approx 34.118 \)
Right side: \( 31.6(1.003)^{25} \approx 34.057 \)
Since \( 34.118 > 34.057 \), the inequality is false.
Answer: FALSE

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

Solution:

The line \( y = 4 \) is a horizontal line situated 4 units above the x-axis, meaning it never intersects the x-axis.
Answer: TRUE

Problem: (True or False)    \( \dfrac{n(n + 3)}{2} = \dfrac{3}{2} \) when \( n = 0 \).

Solution:

Substitute \( n = 0 \) into the expression: \[ \dfrac{0(0 + 3)}{2} = 0 \neq \dfrac{3}{2} \] Answer: FALSE

Problem: (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:

Using the distance formula \( |a - b| = |b - a| \):
Distance 1: \( |-9 - 20| = |-29| = 29 \)
Distance 2: \( |9 - (-20)| = |9 + 20| = 29 \)
Answer: TRUE

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

Solution:

Substitute \( x = -3 \) into the function: \[ f(-3) = \sqrt{1 - (-3)} = \sqrt{1 + 3} = \sqrt{4} = 2 \] Answer: TRUE

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

Solution:

Rewrite the equation in slope-intercept form (\( y = mx + b \)): \[ 2y = -2x + 2 \implies y = -x + 1 \] The slope is \( m = -1 \), not 2.
Answer: FALSE

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

Solution:

Absolute values are non-negative, meaning they can be positive or zero. For instance, when \( x = -5 \), \( |-5 + 5| = 0 \). Because it can equal zero, it is not *always* positive.
Answer: FALSE

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

Solution:

Using the distance formula: \[ \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25} = 5 \] Answer: TRUE

Problem: (True or False)    \( \dfrac{1}{2x - 4} \) is undefined when \( x = -4 \).

Solution:

Substitute \( x = -4 \): \[ \dfrac{1}{2(-4) - 4} = \dfrac{1}{-12} \] This value is defined. The expression is undefined where the denominator equals zero: \[ 2x - 4 = 0 \implies x = 2 \] Answer: FALSE

Problem: (True or False)    \( \left(-\dfrac{1}{5}\right)^{-2} = 25 \)

Solution:

A negative exponent inverts the fraction and squares it: \[ \left(-\dfrac{1}{5}\right)^{-2} = (-5)^2 = 25 \] Answer: TRUE

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

Solution:

The reciprocal of a number \( x \) is \( \dfrac{1}{x} \). Since division by zero is undefined, the reciprocal of 0 does not exist.
Answer: FALSE

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

Solution:

The additive inverse of a number \( a \) is the value that yields 0 when added to \( a \). Since \( -10 + 10 = 0 \), 10 is indeed the additive inverse.
Answer: TRUE

Problem: (True or False)    \( \dfrac{1}{x - 4} = \dfrac{1}{x} - \dfrac{1}{4} \)

Solution:

Test with counterexample \( x = 8 \):
Left side: \( \dfrac{1}{8 - 4} = \dfrac{1}{4} \)
Right side: \( \dfrac{1}{8} - \dfrac{1}{4} = \dfrac{1 - 2}{8} = -\dfrac{1}{8} \)
Since \( \dfrac{1}{4} \neq -\dfrac{1}{8} \), the property is invalid.
Answer: FALSE