Question

warm up 8.1 matching solutions consider the unfinished equation 12x(x - 3) + 18 = ______. match the following expressions with the number of solutions the equation would have with that expression on the right hand side. 1. 6(2x - 3) - one solution 2. 4(3x - 3) - no solutions 3. 4(2x - 3) - all solutions

Explanation:

Step1: Simplify left-hand side

First, expand and combine like terms for $12(x - 3) + 18$:

$$\begin{align*} 12(x - 3) + 18 &= 12x - 36 + 18\\ &= 12x - 18 \end{align*}$$

Step2: Analyze Expression 1

Simplify $6(2x - 3)$:

$$ 6(2x - 3) = 12x - 18 $$

Set equal to left-hand side: $12x - 18 = 12x - 18$. This is always true, so all solutions.

Step3: Analyze Expression 2

Simplify $4(3x - 3)$:

$$ 4(3x - 3) = 12x - 12 $$

Set equal to left-hand side: $12x - 18 = 12x - 12$. Subtract $12x$: $-18 = -12$, which is false, so no solutions.

Step4: Analyze Expression 3

Simplify $4(2x - 3)$:

$$ 4(2x - 3) = 8x - 12 $$

Set equal to left-hand side: $12x - 18 = 8x - 12$. Solve for $x$:

$$ 12x - 8x = -12 + 18\\ 4x = 6\\ x = \frac{3}{2} $$

This has one unique solution.

Answer:

1. $6(2x - 3)$ → All solutions
2. $4(3x - 3)$ → No solutions
3. $4(2x - 3)$ → One solution