\( | 2 x + 7 | - k \leq 5 \)

To solve the inequality: \[ |2x + 7| - k \leq 5 \] **Step 1: Isolate the Absolute Value** First, isolate the absolute value expression by adding \( k \) to both sides: \[ |2x + 7| \leq k + 5 \] **Step 2: Understand the Conditions** For the absolute value inequality \( |A| \leq B \) to have real solutions, the right-hand side \( B \) must be non-negative. Therefore: \[ k + 5 \geq 0 \quad \Rightarrow \quad k \geq -5 \] If \( k + 5 < 0 \), the inequality \( |2x + 7| \leq k + 5 \) has no real solutions because an absolute value cannot be negative. **Step 3: Solve the Absolute Value Inequality** Assuming \( k \geq -5 \), the inequality \( |2x + 7| \leq k + 5 \) can be rewritten without the absolute value as: \[ -(k + 5) \leq 2x + 7 \leq k + 5 \] **Step 4: Solve for \( x \)** 1. **Lower Bound:** \[ -k - 5 \leq 2x + 7 \] Subtract 7 from both sides: \[ -k - 12 \leq 2x \] Divide by 2: \[ -\frac{k}{2} - 6 \leq x \] 2. **Upper Bound:** \[ 2x + 7 \leq k + 5 \] Subtract 7 from both sides: \[ 2x \leq k - 2 \] Divide by 2: \[ x \leq \frac{k}{2} - 1 \] **Step 5: Combine the Results** Putting both bounds together, the solution for \( x \) is: \[ -\frac{k}{2} - 6 \leq x \leq \frac{k}{2} - 1 \] **Summary of Solutions:** - **If \( k \geq -5 \):** \[ -\frac{k}{2} - 6 \leq x \leq \frac{k}{2} - 1 \] - **If \( k < -5 \):** \[ \text{No real solutions} \] **Example:** Suppose \( k = 1 \): 1. Check if \( k \geq -5 \): Yes, since \( 1 \geq -5 \). 2. Compute the bounds: \[ -\frac{1}{2} - 6 = -6.5 \leq x \leq \frac{1}{2} - 1 = -0.5 \] 3. So, \( x \) lies in the interval \([-6.5, -0.5]\).