https://file.aiquickdraw.com/mathbot/file/1727215996430bha9e7xn.jpg

Let's solve the given inequality step by step:

### Given Inequality:
\[
-2(a - 7) - 3a \geq -31
\]

### Step 1: Distribute \(-2\) into \((a - 7)\)
\[
-2 \cdot a + (-2) \cdot (-7) = -2a + 14
\]
Thus, the inequality becomes:
\[
-2a + 14 - 3a \geq -31
\]

### Step 2: Combine like terms
\[
-2a - 3a = -5a
\]
So, the inequality is now:
\[
-5a + 14 \geq -31
\]

### Step 3: Isolate the term with \(a\)
Subtract 14 from both sides:
\[
-5a \geq -31 - 14
\]
\[
-5a \geq -45
\]

### Step 4: Solve for \(a\)
To isolate \(a\), divide both sides of the inequality by \(-5\). **Remember**: When dividing or multiplying an inequality by a negative number, the inequality sign flips.
\[
a \leq \frac{-45}{-5}
\]
\[
a \leq 9
\]

### Final Solution:
\[
a \leq 9
\]

### Conclusion:
The solution to the inequality is \(a \leq 9\).

Would you like more details or have any questions on these steps?

Here are 5 related questions to expand your understanding:
1. What happens to the inequality symbol when dividing by a negative number?
2. How do you check if a value satisfies the inequality?
3. How would the solution change if the inequality was \( > \) instead of \( \geq \)?
4. What are the key differences between solving an equation and an inequality?
5. Can you graph the solution \(a \leq 9\) on a number line?

**Tip:** When solving inequalities, always remember to reverse the inequality symbol when multiplying or dividing by a negative number!

Step-by-step guide to solving the inequality -2(a - 7) - 3a ≥ -31. This problem covers key algebraic concepts such as the distributive property and rules for solving inequalities. Suitable for Grades 7-9.

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