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)
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4(3x-3)
4(2x-3)
6(2x-3)

Answer 1

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

1. Start with the left-hand side: \[ 12(x - 3) + 18 \]
2. Distribute \(12\): \[ 12x - 36 + 18 \]
3. Combine like terms: \[ 12x - 18 \]

So, the equation simplifies to: \[ 12x - 18 = \text{RHS} \]

Next, we can equate this to each of the suggested expressions (RHS) and analyze the resulting equations.

1. For \(4(3x - 3)\): \[ 12x - 18 = 4(3x - 3) \] Simplifying the right-hand side: \[ 4(3x - 3) = 12x - 12 \] Now we have: \[ 12x - 18 = 12x - 12 \] Subtract \(12x\) from both sides: \[ -18 = -12 \] This is a contradiction, meaning there are no solutions.

2. For \(4(2x - 3)\): \[ 12x - 18 = 4(2x - 3) \] Simplifying the right-hand side: \[ 4(2x - 3) = 8x - 12 \] Now we have: \[ 12x - 18 = 8x - 12 \] Subtract \(8x\) from both sides: \[ 4x - 18 = -12 \] Adding \(18\) to both sides gives: \[ 4x = 6 \] Therefore: \[ x = \frac{3}{2} \] This means there is one solution.

3. For \(6(2x - 3)\): \[ 12x - 18 = 6(2x - 3) \] Simplifying the right-hand side: \[ 6(2x - 3) = 12x - 18 \] Now we have: \[ 12x - 18 = 12x - 18 \] This is an identity, meaning there are infinitely many solutions.

Now we can match the expressions to the number of solutions:

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

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