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Let's solve the inequality:

\[
-7x + 10 \geq 8(4 - x)
\]

### Step 1: Distribute on the right-hand side
We distribute the 8 over \(4 - x\):

\[
-7x + 10 \geq 8 \cdot 4 - 8 \cdot x
\]
\[
-7x + 10 \geq 32 - 8x
\]

### Step 2: Move all terms involving \(x\) to one side
Add \(8x\) to both sides to eliminate \(x\) from the right-hand side:

\[
-7x + 8x + 10 \geq 32
\]
\[
x + 10 \geq 32
\]

### Step 3: Isolate \(x\)
Subtract 10 from both sides:

\[
x \geq 32 - 10
\]
\[
x \geq 22
\]

### Final Answer
The solution is \(x \geq 22\). Therefore, the correct answer is **C**.

Would you like further clarification or details on any step?

### Related Questions:
1. How do you solve linear inequalities involving distribution?
2. What happens if you divide or multiply both sides of an inequality by a negative number?
3. How would the inequality change if we had a strict inequality instead of a "greater than or equal to"?
4. How do you check if your solution to an inequality is correct?
5. What are the common mistakes when solving inequalities?

**Tip:** Always remember to flip the inequality sign when you multiply or divide both sides of an inequality by a negative number.

Which of the following inequalities describes all of the solutions to -7x + 10 ≥ 8(4 - x)?

Learn how to solve the linear inequality -7x + 10 ≥ 8(4 - x) step by step. This problem involves key algebraic concepts, including distributing, combining like terms, and isolating the variable. Suitable for students in Grades 7-10.

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