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The given inequality is:

\[
3 - 8|6 - 7m| \geq -61
\]

To solve this inequality, let's follow these steps:

### Step 1: Isolate the absolute value expression
Subtract 3 from both sides:

\[
-8|6 - 7m| \geq -61 - 3
\]

\[
-8|6 - 7m| \geq -64
\]

### Step 2: Divide by -8 and reverse the inequality sign
Since we're dividing by a negative number, we need to reverse the inequality:

\[
|6 - 7m| \leq 8
\]

### Step 3: Break down the absolute value inequality
The inequality \( |6 - 7m| \leq 8 \) can be split into two cases:

1. \( 6 - 7m \leq 8 \)
2. \( 6 - 7m \geq -8 \)

### Step 4: Solve each inequality
**Case 1:**

\[
6 - 7m \leq 8
\]

Subtract 6 from both sides:

\[
-7m \leq 2
\]

Divide by -7 (and reverse the inequality sign):

\[
m \geq -\frac{2}{7}
\]

**Case 2:**

\[
6 - 7m \geq -8
\]

Subtract 6 from both sides:

\[
-7m \geq -14
\]

Divide by -7 (and reverse the inequality sign):

\[
m \leq 2
\]

### Step 5: Combine the solutions
The solution to the inequality is:

\[
-\frac{2}{7} \leq m \leq 2
\]

This is the range of values that satisfy the original inequality.

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. How does reversing the inequality sign affect the solution when dividing by a negative number?
2. What are the key steps in solving inequalities involving absolute values?
3. Can you graph the solution set for \( -\frac{2}{7} \leq m \leq 2 \)?
4. How would the solution change if the original inequality were \( \geq -60 \) instead of \( \geq -61 \)?
5. What is the significance of combining the two cases when solving absolute value inequalities?

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

Learn how to solve the absolute value inequality 3 - 8|6 - 7m| ≥ -61 step-by-step. This solution covers algebraic concepts, including isolating the absolute value, reversing the inequality when dividing by a negative number, and combining the solutions for different cases. Suitable for students in Grades 8-10.

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