Question

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. (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(2x-3) 4(3x-3)

Answer 1

To determine how many solutions the equation \(12(x - 3) + 18 = \text{RHS}\) has for each given expression on the right-hand side (RHS), we need to simplify the left-hand side (LHS) and then compare it to the expressions.

1. First, simplify the LHS:

\[ 12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18 \]

So the equation we are working with is:

\[ 12x - 18 = \text{RHS} \]

2. Now, let's consider each expression for the RHS:

• For \(6(2x-3)\):

\[ 6(2x - 3) = 12x - 18 \]

This means \(12x - 18 = 12x - 18\), which is an identity. It has infinite solutions.

• For \(4(2x-3)\):

\[ 4(2x - 3) = 8x - 12 \]

So, we have \(12x - 18 = 8x - 12\). Solving for \(x\):

\[ 12x - 8x = -12 + 18 \implies 4x = 6 \implies x = \frac{3}{2} \]

Thus, there is one solution.

• For \(4(3x-3)\):

\[ 4(3x - 3) = 12x - 12 \]

So, we have \(12x - 18 = 12x - 12\). This simplifies to:

\[ -18 = -12 \]

This is a contradiction, meaning there are no solutions.

3. Now we can match each expression with the corresponding number of solutions:

• \(6(2x-3)\) → Infinite Solutions
• \(4(2x-3)\) → One Solution
• \(4(3x-3)\) → No Solutions

You can fill in the appropriate options based on this reasoning.