6. For each equation, determine whether it has no solutions, one solution, or infinite solutions. If it has one solution, solve to find the value of \( x \) that makes the statement true. Show/explain your reasoning \( \begin{array}{ll}\text { a. } x-4(3-2 x)=3(3 x+4) & \text { b. } x-4(3+2 x)=3(3 x+4) \\ \text { c. } x-4(3-2 x)=3(3 x-4) & \end{array} \)

To determine whether each equation has no solutions, one solution, or infinite solutions, we will solve each equation step by step. ### a. \( x - 4(3 - 2x) = 3(3x + 4) \) 1. **Distribute** on both sides: \[ x - 12 + 8x = 9x + 12 \] This simplifies to: \[ 9x - 12 = 9x + 12 \] 2. **Rearranging** the equation: \[ 9x - 9x - 12 = 12 \] This simplifies to: \[ -12 = 12 \] Since this statement is false, the equation has **no solutions**. ### b. \( x - 4(3 + 2x) = 3(3x + 4) \) 1. **Distribute** on both sides: \[ x - 12 - 8x = 9x + 12 \] This simplifies to: \[ -7x - 12 = 9x + 12 \] 2. **Rearranging** the equation: \[ -7x - 9x = 12 + 12 \] This simplifies to: \[ -16x = 24 \] 3. **Solving for \( x \)**: \[ x = -\frac{24}{16} = -\frac{3}{2} \] Thus, this equation has **one solution**, \( x = -\frac{3}{2} \). ### c. \( x - 4(3 - 2x) = 3(3x - 4) \) 1. **Distribute** on both sides: \[ x - 12 + 8x = 9x - 12 \] This simplifies to: \[ 9x - 12 = 9x - 12 \] 2. **Rearranging** the equation: \[ 9x - 9x - 12 = -12 \] This simplifies to: \[ -12 = -12 \] Since this statement is always true, the equation has **infinite solutions**. ### Summary of Results: - **a.** No solutions - **b.** One solution: \( x = -\frac{3}{2} \) - **c.** Infinite solutions