To solve the inequalities \(5 + x \leq 8\) and \(13 + x \geq 7\), we will break down each inequality step-by-step to find the range of values for \(x\). ### Step 1: Solve the first inequality **Inequality:** \[ 5 + x \leq 8 \] **Step to isolate \(x\):** 1. Subtract 5 from both sides: \[ x \leq 8 - 5 \] 2. Simplifying gives: \[ x \leq 3 \] ### Step 2: Solve the second inequality **Inequality:** \[ 13 + x \geq 7 \] **Step to isolate \(x\):** 1. Subtract 13 from both sides: \[ x \geq 7 - 13 \] 2. Simplifying gives: \[ x \geq -6 \] ### Step 3: Combine the results Now we have two inequalities: 1. \(x \leq 3\) 2. \(x \geq -6\) Combining these gives us the range: \[ -6 \leq x \leq 3 \] ### Conclusion The range of \(x\) that satisfies both inequalities is: \[ -6 \leq x \leq 3 \] ### Final Answer The correct option is **C. -6 \leq x \leq 3**. ### Explanation of Other Options - **Option A: -3 ≤ x ≤ 3** This option is incorrect because it restricts the lower bound of \(x\) to -3, which does not satisfy the second inequality \(x \geq -6\). The correct lower bound is -6. - **Option B: 3 ≤ x ≤ 6** This option is incorrect because it suggests that \(x\) must be at least 3, which contradicts the first inequality \(x \leq 3\). Therefore, \(x\) cannot be both greater than or equal to 3 and less than or equal to 3 simultaneously. - **Option D: -6 ≤ x ≤ -3** This option is incorrect because it restricts the upper bound of \(x\) to -3, which does not satisfy the first inequality \(x \leq 3\). The correct upper bound is 3. ### Common Pitfalls 1. **Misreading the inequalities:** Ensure you correctly interpret the symbols \( \leq \) and \( \geq \). 2. **Combining ranges incorrectly:** When combining inequalities, ensure that both conditions are satisfied simultaneously. 3. **Arithmetic errors:** Double-check calculations when isolating \(x\) to avoid mistakes. ### Revision Summary - Solve each inequality separately to isolate \(x\). - Combine the results to find the overall range of \(x\). - Check each option against the derived range to identify the correct answer. - Be cautious of common pitfalls in interpreting inequalities and combining results.