The solution set of the equation `|2x+3| -|x-1|=6` is :

Options

A. `x in (-10 , 2)`
B. `x in [-10,2)`
C. `x in [-10,2]`
D. `x in {-10,2}`

Answer

To solve the equation \( |2x + 3| - |x - 1| = 6 \), we will analyze the expression by considering different cases based on the critical points where the expressions inside the absolute values change sign. The critical points are \( x = -\frac{3}{2} \) and \( x = 1 \).

### Step 1: Identify the critical points
The critical points are found by setting the expressions inside the absolute values to zero:
- \( 2x + 3 = 0 \) gives \( x = -\frac{3}{2} \)
- \( x - 1 = 0 \) gives \( x = 1 \)

These points divide the number line into three intervals:
1. \( (-\infty, -\frac{3}{2}) \)
2. \( [-\frac{3}{2}, 1) \)
3. \( [1, \infty) \)

### Step 2: Analyze each interval

#### Case 1: \( x \geq 1 \)
In this interval, both expressions inside the absolute values are positive:
\[ |2x + 3| = 2x + 3 \quad \text{and} \quad |x - 1| = x - 1 \]
Substituting these into the equation:
\[ 2x + 3 - (x - 1) = 6 \]
Simplifying:
\[ 2x + 3 - x + 1 = 6 \implies x + 4 = 6 \implies x = 2 \]
Since \( x = 2 \) is within the interval \( [1, \infty) \), it is a valid solution.

#### Case 2: \( -\frac{3}{2} \leq x < 1 \)
In this interval, \( 2x + 3 \) is positive and \( x - 1 \) is negative:
\[ |2x + 3| = 2x + 3 \quad \text{and} \quad |x - 1| = -(x - 1) = -x + 1 \]
Substituting these into the equation:
\[ 2x + 3 - (-x + 1) = 6 \]
Simplifying:
\[ 2x + 3 + x - 1 = 6 \implies 3x + 2 = 6 \implies 3x = 4 \implies x = \frac{4}{3} \]
However, \( \frac{4}{3} \) is greater than 1, so it does not lie in this interval. Thus, there are no valid solutions from this case.

#### Case 3: \( x < -\frac{3}{2} \)
In this interval, both expressions inside the absolute values are negative:
\[ |2x + 3| = -(2x + 3) = -2x - 3 \quad \text{and} \quad |x - 1| = -(x - 1) = -x + 1 \]
Substituting these into the equation:
\[ -2x - 3 - (-x + 1) = 6 \]
Simplifying:
\[ -2x - 3 + x - 1 = 6 \implies -x - 4 = 6 \implies -x = 10 \implies x = -10 \]
Since \( -10 < -\frac{3}{2} \), this is a valid solution.

### Step 3: Compile the solutions
The valid solutions we found are:
1. \( x = 2 \)
2. \( x = -10 \)

### Final Solution Set
The solution set of the equation \( |2x + 3| - |x - 1| = 6 \) is:
\[ \{ -10, 2 \} \]