To find the solution set for the inequality \(|m+2| + 7 \leq 16\), we can proceed as follows:

### Step-by-step Solution

1. **Isolate the absolute value term:**

   Start with the given inequality:
   \[
   |m+2| + 7 \leq 16
   \]

   Subtract 7 from both sides to isolate the absolute value expression:
   \[
   |m+2| \leq 9
   \]

2. **Interpret the absolute value inequality:**

   The inequality \(|m+2| \leq 9\) means that the quantity inside the absolute value, \(m+2\), is between \(-9\) and \(9\). This can be written as a compound inequality:
   \[
   -9 \leq m+2 \leq 9
   \]

3. **Solve the compound inequality for \(m\):**

   Subtract 2 from each part of the inequality to solve for \(m\):
   \[
   -9 - 2 \leq m + 2 - 2 \leq 9 - 2
   \]
   Simplifying this gives:
   \[
   -11 \leq m \leq 7
   \]

### Solution Set

The solution set is all values of \(m\) such that:
\[
m \in [-11, 7]
\]

This means \(m\) can be any number between \(-11\) and \(7\), inclusive of both endpoints.

### Interval Notation

In interval notation, the solution set is written as:
\[
[-11, 7]
\]

Would you like more details or have any questions?

### Related Questions:

1. How would the solution change if the inequality was \(|m+2| + 7 < 16\)?
2. What happens if we change the sign of the inequality to \(|m+2| + 7 \geq 16\)?
3. How can we graph the solution \([-11, 7]\) on a number line?
4. How would you solve the inequality \(|2m - 5| + 3 \leq 12\)?
5. What are the steps to solve an absolute value equation like \(|m+2| = 9\)?

### Tip:

When solving absolute value inequalities, isolate the absolute value term first, then translate the inequality into a compound inequality. This approach helps in solving these inequalities systematically and finding the solution set.

Learn how to solve the inequality |m+2| + 7 ≤ 16 with a detailed step-by-step solution. Understand how to handle absolute value inequalities and find the solution set [-11, 7]. Suitable for students in Grades 7-9.

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