https://file.aiquickdraw.com/mathbot/file/1727429688576ufzxk79x.jpg

Let's solve the inequalities from the image one by one.

### Problem 13: 
Solve the following inequalities and combine the solutions:
- \(-6 + 2a \geq 22\)
- \(10 + 3a \leq 22\)

**Solution**:

1. **First inequality**: 
   \[
   -6 + 2a \geq 22
   \]
   Add 6 to both sides:
   \[
   2a \geq 28
   \]
   Divide by 2:
   \[
   a \geq 14
   \]

2. **Second inequality**:
   \[
   10 + 3a \leq 22
   \]
   Subtract 10 from both sides:
   \[
   3a \leq 12
   \]
   Divide by 3:
   \[
   a \leq 4
   \]

Now, combine the solutions. Since the two inequalities are connected by "OR", the solution is any \(a\) that satisfies either \(a \geq 14\) or \(a \leq 4\). Therefore, the solution set is:
\[
a \leq 4 \text{ or } a \geq 14
\]
Graphically, the solution is two separate intervals, one starting at negative infinity to 4 and the other starting at 14 to positive infinity.

---

### Problem 14:
Solve the following inequalities:
- \(4n - 5 \geq 23\)
- \(n - 2 \leq 1\)

**Solution**:

1. **First inequality**:
   \[
   4n - 5 \geq 23
   \]
   Add 5 to both sides:
   \[
   4n \geq 28
   \]
   Divide by 4:
   \[
   n \geq 7
   \]

2. **Second inequality**:
   \[
   n - 2 \leq 1
   \]
   Add 2 to both sides:
   \[
   n \leq 3
   \]

Combine the results. Since the inequalities are connected by "OR", the solution is \(n \geq 7\) or \(n \leq 3\). The solution set is:
\[
n \leq 3 \text{ or } n \geq 7
\]
Graphically, it is similar to problem 13 with two intervals, one from negative infinity to 3 and the other from 7 to positive infinity.

---

Would you like to see the solutions for problems 15 and 16, or do you need further clarification on the above steps?

### Related questions:
1. How do you solve compound inequalities connected by "OR" or "AND"?
2. What does the solution set of an "OR" inequality look like graphically?
3. How can you verify solutions for inequalities?
4. How do we graph the solution of inequalities on a number line?
5. What happens when both inequalities have no overlap in an "AND" compound inequality?

### Tip:
When solving inequalities, always remember to flip the inequality sign if you multiply or divide both sides by a negative number.

Solve the inequalities -6 + 2a ≥ 22 OR 10 + 3a ≤ 22, and 4n - 5 ≥ 23 OR n - 2 ≤ 1.

Learn how to solve compound inequalities such as -6 + 2a ≥ 22 or 10 + 3a ≤ 22, and how to combine solutions in compound inequalities. This solution demonstrates step-by-step algebraic methods to solve inequalities and find the solution set, suitable for Grades 8-10.

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