Question

solve.\\(|3x + 2| = |x - 5|\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\a. the solution set is \\(\boxed{}\\). (type an integer or a simplified fraction. use a comma to separate answers as needed.)\\b. the solution is all real numbers.

Explanation:

Step1: Use absolute value property

If \(|a| = |b|\), then \(a = b\) or \(a = -b\). So we have two cases:
Case 1: \(3x + 2 = x - 5\)
Case 2: \(3x + 2 = -(x - 5)\)

Step2: Solve Case 1

\(3x + 2 = x - 5\)
Subtract \(x\) from both sides: \(2x + 2 = -5\)
Subtract 2 from both sides: \(2x = -7\)
Divide by 2: \(x = -\frac{7}{2}\)

Step3: Solve Case 2

\(3x + 2 = -x + 5\)
Add \(x\) to both sides: \(4x + 2 = 5\)
Subtract 2 from both sides: \(4x = 3\)
Divide by 4: \(x = \frac{3}{4}\)

Answer:

\(-\frac{7}{2}, \frac{3}{4}\)