Consider the unfinished equation 12(x - 3) + 18 = _______. Match the following expressions with the number of solutions the equation would have with that expression on the right-hand side.

Question

(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
6(2x-3)
4(3x-3)
4(2x-3)

Answer 1

To determine the number of solutions for the equation \( 12(x - 3) + 18 = \text{(expression)} \), we first simplify the left-hand side:

1. Expand \( 12(x - 3) + 18 \): \[ 12x - 36 + 18 = 12x - 18 \]

Now, we rewrite the equation: \[ 12x - 18 = \text{(expression)} \]

Next, we will analyze each expression to see how many solutions exist based on equality with \( 12x - 18 \).

1. Expression: \( 6(2x - 3) \)

\[ 6(2x - 3) = 12x - 18 \] Which means: \[ 12x - 18 = 12x - 18 \quad \text{(infinitely many solutions)} \]

2. Expression: \( 4(3x - 3) \)

\[ 4(3x - 3) = 12x - 12 \] Set it equal: \[ 12x - 18 = 12x - 12 \implies -18 = -12 \quad \text{(no solution)} \]

3. Expression: \( 4(2x - 3) \)

\[ 4(2x - 3) = 8x - 12 \] Set it equal: \[ 12x - 18 = 8x - 12 \implies 4x = 6 \implies x = \frac{3}{2} \quad \text{(one solution)} \]

Summary of Solutions

• \( 6(2x - 3) \): Infinitely many solutions
• \( 4(3x - 3) \): No solutions
• \( 4(2x - 3) \): One solution

Matching Responses

Now, we can match the expressions with the number of solutions:

1. \( 6(2x - 3) \) → Infinitely many solutions
2. \( 4(3x - 3) \) → No solutions
3. \( 4(2x - 3) \) → One solution

Input responses:

• Infinitely many solutions: 6(2x - 3)
• No solutions: 4(3x - 3)
• One solution: 4(2x - 3)

Please place the expressions in the appropriate categories based on the number of solutions.