Question

Solve the inequality. You must show ALL work to earn full credit.

-2(a-7)-3m>_-31

Answer 1

To solve the inequality \(-2(a-7)-3m \geq -31\), we will follow these steps.

### Step 1: Distribute the \(-2\)
We start by distributing the \(-2\) across the expression \(a-7\).

\[
-2(a-7) = -2a + 14
\]

Now we can rewrite the inequality:

\[
-2a + 14 - 3m \geq -31
\]

### Step 2: Combine like terms
Next, we can isolate the terms on the left side. We will first subtract \(14\) from both sides of the inequality:

\[
-2a - 3m \geq -31 - 14
\]

Calculating the right side gives us:

\[
-31 - 14 = -45
\]

So we have:

\[
-2a - 3m \geq -45
\]

### Step 3: Isolate \( -3m \)
Now, let's isolate the term \(-3m\) by adding \(2a\) to both sides:

\[
-3m \geq 2a - 45
\]

### Step 4: Solve for \(m\)
Next, we solve for \(m\) by dividing both sides by \(-3\). Remember that dividing by a negative number reverses the inequality sign:

\[
m \leq \frac{2a - 45}{-3}
\]

This can also be rewritten as:

\[
m \leq -\frac{2a}{3} + 15
\]

### Conclusion
The solution to the inequality \(-2(a-7) - 3m \geq -31\) is:

\[
m \leq -\frac{2a}{3} + 15
\]

This indicates that the value of \(m\) is less than or equal to \(-\frac{2}{3}a + 15\).