Intermediate Algebra Practice
True or False Algebra Questions With Solutions

Problem: (True or False)   \( x^2 \) and \( 2x \) are like terms.

Solution:

The statement is FALSE because the two terms do not have the same exponent on \( x \).

Problem: (True or False)   \( x^{-3} \) and \( -3x \) are unlike terms.

Solution:

The statement is TRUE because the terms have different exponents (powers of \( x \)).

Problem: (True or False)   \( \dfrac{1}{x - 9} = 0 \) for \( x = 9 \).

Solution:

Substitute \( x = 9 \) into the expression: \[ \dfrac{1}{9 - 9} = \dfrac{1}{0} = \text{undefined} \] The statement is FALSE.

Problem: (True or False)   The set of ordered pairs \( \{(0, 0), (2, 0), (3, 0), (10, 0)\} \) represents a function.

Solution:

All input values (\( x \)-coordinates) are distinct, meaning each input maps to exactly one output. The statement is TRUE.

Problem: (True or False)   \( |a - b| = b - a \) if \( b - a < 0 \).

Solution:

Recall that for any real number \( x \), if \( x < 0 \), then \( |x| = -x \). If \( b - a < 0 \), then \( a - b > 0 \), and: \[ |a - b| = a - b \] The statement is FALSE.

Problem: (True or False)   \( |x^2 + 1| = x^2 + 1 \).

Solution:

Since \( x^2 + 1 \) is strictly positive for all real numbers \( x \), its absolute value is equal to itself. The statement is TRUE.

Problem: (True or False)   \( \sqrt{(x - 5)^2} = x - 5 \).

Solution:

Let \( x = -4 \).
Left-hand side: \( \sqrt{(-4 - 5)^2} = \sqrt{(-9)^2} = \sqrt{81} = 9 \)
Right-hand side: \( -4 - 5 = -9 \)
Since \( 9 \neq -9 \), the statement is FALSE (the correct simplification is \( \sqrt{(x - 5)^2} = |x - 5| \)).

Problem: (True or False)   \( (x - 2)(x + 2) = x^2 - 4x - 4 \).

Solution:

Expand using the difference of squares: \[ (x - 2)(x + 2) = x^2 + 2x - 2x - 4 = x^2 - 4 \] The statement is FALSE.

Problem: (True or False)   \( \sqrt{x + 9} = \sqrt{x} + \sqrt{9} \), for all \( x \ge 0 \).

Solution:

Let \( x = 16 \).
Left-hand side: \( \sqrt{16 + 9} = \sqrt{25} = 5 \)
Right-hand side: \( \sqrt{16} + \sqrt{9} = 4 + 3 = 7 \)
Since \( 5 \neq 7 \), the statement is FALSE.

Problem: (True or False)   \( |x - 3| = |x| + |3| \), for all negative real numbers \( x \).

Solution:

Since \( x < 0 \), both \( x \) and \( x - 3 \) are negative. Thus: \[ |x - 3| = -(x - 3) = 3 - x = |3| + |x| \] The statement is TRUE.

Problem: (True or False)   \( (x + 2)^3 = x^3 + 2^3 \), for all real \( x \).

Solution:

Expanding the binomial cube yields: \[ (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \] Since this differs from \( x^3 + 8 \), the statement is FALSE.

Problem: (True or False)   If \( k = 4 \), then the equation \( x^2 - kx = -4 \) has one solution only.

Solution:

Substitute \( k = 4 \): \[ x^2 - 4x = -4 \implies x^2 - 4x + 4 = 0 \implies (x - 2)^2 = 0 \] This yields a single solution: \( x = 2 \). The statement is TRUE.

Problem: (True or False)   The discriminant of the equation \( 2x^2 - 4x + 9 = 0 \) is negative.

Solution:

Calculate the discriminant \( \Delta = b^2 - 4ac \): \[ \Delta = (-4)^2 - 4(2)(9) = 16 - 72 = -56 \] Since \(-56 < 0\), the discriminant is indeed negative. The statement is TRUE.

Problem: (True or False)   The degree of the polynomial \( P(x) = (x - 2)(-x + 3)(x - 4) \) is equal to \(-3\).

Solution:

Multiplying the leading terms gives \( (x)(-x)(x) = -x^3 \), meaning the degree of the polynomial is 3, not \(-3\). The statement is FALSE.

Problem: (True or False)   The distance between the points \( (0, 0) \) and \( (1, 1) \) in a rectangular coordinate system is equal to 1.

Solution:

Using the distance formula: \[ \text{Distance} = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \] Since \( \sqrt{2} \neq 1 \), the statement is FALSE.

Problem: (True or False)   The slope of the line \( 2x + 3y = -2 \) is negative.

Solution:

Rewrite in slope-intercept form: \[ 3y = -2x - 2 \implies y = -\dfrac{2}{3}x - \dfrac{2}{3} \] The slope is \( -\dfrac{2}{3} \), which is negative. The statement is TRUE.

Problem: (True or False)   The relation \( 2y + x^2 = 2 \) represents \( y \) as a function of \( x \).

Solution:

Solve for \( y \): \[ 2y = -x^2 + 2 \implies y = -\dfrac{1}{2}x^2 + 1 \] Since each input \( x \) yields exactly one output \( y \), it represents a function. The statement is TRUE.

Problem: (True or False)   The relation \( 2y + x^2 = 2 \) represents \( x \) as a function of \( y \).

Solution:

Solve for \( x \): \[ x^2 = 2 - 2y \implies x = \pm\sqrt{2 - 2y} \] Since a single \( y \) value can produce two different \( x \) values, it does not represent \( x \) as a function of \( y \). The statement is FALSE.

Problem: (True or False)   The relation \( |x| = |y| \) does NOT represent \( x \) as a function of \( y \).

Solution:

Solve for \( x \): \[ |x| = |y| \implies x = \pm y \] Because one \( y \) value can yield two different \( x \) values, \( x \) is not a function of \( y \). Therefore, the statement that it *does not* represent a function is TRUE.

Problem: (True or False)   The relation \( |x| = |y| \) does NOT represent \( y \) as a function of \( x \).

Solution:

Solve for \( y \): \[ |x| = |y| \implies y = \pm x \] Because one \( x \) value can yield two different \( y \) values, \( y \) is not a function of \( x \). Therefore, the statement that it *does not* represent a function is TRUE.