Question

solve the equation.\\(|2x - 3| = 6\\)\\(\\)\\(\\)select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\\)\\(\\)a. the solution set is \\(\\{\\ \\}\\).\\(\\)(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)\\(\\)\\(\\)b. the solution is all real numbers.\\(\\)\\(\\)c. the solution is the empty set.

Explanation:

Step1: Recall absolute value definition

For \(|A| = B\) (\(B\geq0\)), \(A = B\) or \(A=-B\). So for \(|2x - 3|=6\), we have two cases:
Case 1: \(2x - 3 = 6\)
Case 2: \(2x - 3=-6\)

Step2: Solve Case 1

Solve \(2x - 3 = 6\). Add 3 to both sides:
\(2x=6 + 3\)
\(2x=9\)
Divide both sides by 2:
\(x=\frac{9}{2}\)

Step3: Solve Case 2

Solve \(2x - 3=-6\). Add 3 to both sides:
\(2x=-6 + 3\)
\(2x=-3\)
Divide both sides by 2:
\(x=-\frac{3}{2}\)

Answer:

The solution set is \(\frac{9}{2},-\frac{3}{2}\) (so the correct choice is A with solution set \(\frac{9}{2},-\frac{3}{2}\))