Question

given the equation ax + b = c, solve for x. describe each statement in parts a to c below as always, sometimes, or never true. explain your answer. b. if a, b, and c are integers, x is an integer. choose the correct answer below. a. the statement is sometimes true because the solution could also be a fraction. b. the statement is always true. the solution for x is an expression involving addition, subtraction, multiplication, and/or division, and these operations always give integers when a, b, and c are integers. c. the statement is sometimes true because the solution could also be a negative number. d. the statement is never true because the solution involves division, meaning, x will be a fraction.

Explanation:

Part 1: Solve \( ax + b = c \) for \( x \)

Step 1: Isolate the term with \( x \)

Subtract \( b \) from both sides of the equation:
\( ax + b - b = c - b \)
Simplifies to: \( ax = c - b \)

Step 2: Solve for \( x \)

Divide both sides by \( a \) (assuming \( a
eq 0 \)):
\( x = \frac{c - b}{a} \)

• Option A: The solution \( x = \frac{c - b}{a} \) can be a fraction (e.g., \( a = 2, b = 1, c = 2 \) gives \( x = \frac{1}{2} \)) or an integer (e.g., \( a = 1, b = 1, c = 3 \) gives \( x = 2 \)). So it is sometimes true that \( x \) is a fraction.
• Option B: Division of integers does not always yield an integer (e.g., \( a = 2, b = 1, c = 2 \) gives \( x = \frac{1}{2} \), not an integer). So the statement "always true" is false.
• Option C: The solution can be negative (e.g., \( a = 1, b = 3, c = 1 \) gives \( x = -2 \)) or positive, and also a fraction. The option incorrectly restricts to "not a negative number," so it is false.
• Option D: The solution can be an integer (e.g., \( a = 1, b = 1, c = 3 \) gives \( x = 2 \)), so "never true" is false.

Answer:

\( x = \frac{c - b}{a} \) (for \( a
eq 0 \))

Part b: Analyze the statement about integers \( a, b, c \) and integer \( x \)