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 \( \begin{array}{ll}\text { a. } x-4(3-2 x)=3(3 x+4) & \text { b. } x-4(3+2 x)=3(3 x+4) \\ \text { c. } x-4(3-2 x)=3(3 x-4) & \end{array} \)
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Alright, let's break down these equations one by one and see what we discover! **a.** Start with the equation: \( x - 4(3 - 2x) = 3(3x + 4) \) Distributing on both sides gives: \( x - 12 + 8x = 9x + 12 \) Combine like terms: \( 9x - 12 = 9x + 12 \) Subtract \( 9x \) from both sides: \( -12 = 12 \) This is a contradiction, which means the equation has **no solutions**. **b.** Now for the second equation: \( x - 4(3 + 2x) = 3(3x + 4) \) Distributing gives: \( x - 12 - 8x = 9x + 12 \) Combine like terms: \( -7x - 12 = 9x + 12 \) Rearranging yields: \( -7x - 9x = 12 + 12 \) Thus, \( -16x = 24 \), so dividing both sides by -16 gives: \( x = -\frac{3}{2} \). There is **one solution**: \( x = -\frac{3}{2} \). **c.** Finally, for the last equation: \( x - 4(3 - 2x) = 3(3x - 4) \) Distributing gives: \( x - 12 + 8x = 9x - 12 \) Combine like terms: \( 9x - 12 = 9x - 12 \) This is an identity, meaning it holds true for all values of \( x \), so there are **infinite solutions**. In summary: - a: No solutions - b: One solution, \( x = -\frac{3}{2} \) - c: Infinite solutions Equations can really put on a show! Always remember to keep an eye out for contradictions or identities that reveal their secrets. Happy solving!
