True/False Algebra Questions with Answers and Explanations

Problem: (True or False)   \(((2^2)^3) = 2^5\)

Solution:

Evaluate using the power of a power property \((a^m)^n = a^{m \cdot n}\): \[ ((2^2)^3) = 2^{2 \cdot 3} = 2^6 = 64 \] Comparing this with \(2^5 = 32\), we see that \(64 \neq 32\).
Answer: FALSE

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

Solution:

Consider two points on a vertical line, such as \((2, 5)\) and \((2, 7)\). Calculating the slope: \[ m = \dfrac{7 - 5}{2 - 2} = \dfrac{2}{0} \] Division by zero is undefined.
Answer: TRUE

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

Solution:

Perpendicular lines must have slopes whose product is \(-1\) (\(m_1 \cdot m_2 = -1\)). If both slopes are positive, their product must be positive, which can never equal \(-1\).
Answer: FALSE

Problem: (True or False)   \(5 > 10\) or \(5 < 7\)

Solution:

The statement \(5 > 10\) is FALSE, but \(5 < 7\) is TRUE. In logic, a compound "or" statement is true if at least one of the conditions is true.
Answer: TRUE

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

Solution:

The input \(6\) is paired with two different outputs (\(4\) and \(-4\)). Because a single input maps to multiple outputs, this relation is not a function.
Answer: FALSE

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

Solution:

The additive inverse of a number \(x\) is \(-x\). For \(-10\): \[ -(-10) = 10 \] Answer: TRUE

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

Solution:

The product of any two positive numbers is always positive.
Answer: FALSE

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

Solution:

The presence of an equals sign (\(=\)) makes it an equation, not an inequality.
Answer: FALSE

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

Solution:

Factoring out the common factor uses the distributive property, not the associative property.
Answer: FALSE

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

Solution:

By definition, the absolute value of any real number is always non-negative (positive or zero).
Answer: FALSE

Problem: (True or False)   \(-2^3 = (-2)^3\)

Solution:

Evaluate both sides:
\(-2^3 = -(2^3) = -8\)
\((-2)^3 = (-2)(-2)(-2) = -8\)
Both expressions equal \(-8\).
Answer: TRUE

Problem: (True or False)   30% of \(x\) is equal to \(0.03x\)

Solution:

30% of \(x\) is written as \(\dfrac{30}{100}x = 0.3x\), not \(0.03x\).
Answer: FALSE

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

Solution:

"At most" translates to less than or equal to (\(\leq\)), so it should be written as \(x \leq 9\).
Answer: FALSE

Problem: (True or False)   \(3^{20} + 3^{20} + 3^{20} = 3^{21}\)

Solution:

Factor out the common term: \[ 3^{20} + 3^{20} + 3^{20} = 3 \cdot 3^{20} = 3^1 \cdot 3^{20} = 3^{21} \] Answer: TRUE

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

Solution:

Converting \(1.5 \times 10^{-5}\) yields \(0.000015\), not \(0.0000015\) (which is \(1.5 \times 10^{-6}\)).
Answer: FALSE

Problem: (True or False)   \(\dfrac{1000}{0} = 0\)

Solution:

Division by zero is undefined in mathematics.
Answer: FALSE

Problem: (True or False)   \(\dfrac{0}{1000} = 0\)

Solution:

Zero divided by any non-zero number is always zero.
Answer: TRUE

Problem: (True or False)   \(0.0000001^0 = 1\)

Solution:

Any non-zero real number raised to the power of 0 equals 1.
Answer: TRUE

Problem: (True or False)   \(\dfrac{1}{(-2)^{-4}} = 16\)

Solution:

Using the negative exponent rule \(\dfrac{1}{a^{-n}} = a^n\): \[ \dfrac{1}{(-2)^{-4}} = (-2)^4 = 16 \] Answer: TRUE

Problem: (True or False)   \(x = 7\) DOES NOT satisfy the inequality \(x - 7 < 0\)

Solution:

Substitute \(x = 7\) into \(x - 7 < 0\): \[ 7 - 7 < 0 \implies 0 < 0 \] This statement is false, meaning \(x = 7\) does not satisfy the inequality. Therefore, the assertion that it "DOES NOT satisfy" is correct.
Answer: TRUE