Let's solve the inequality step by step.

Given inequality:
\[
8(6x + 5) + 2 \ge 2x + 7
\]

### Step 1: Distribute the 8 on the left-hand side.
\[
8 \times 6x + 8 \times 5 + 2 \ge 2x + 7
\]
\[
48x + 40 + 2 \ge 2x + 7
\]

### Step 2: Combine like terms on the left-hand side.
\[
48x + 42 \ge 2x + 7
\]

### Step 3: Subtract \(2x\) from both sides to move the terms involving \(x\) to one side.
\[
48x - 2x + 42 \ge 7
\]
\[
46x + 42 \ge 7
\]

### Step 4: Subtract 42 from both sides to isolate the term with \(x\).
\[
46x \ge 7 - 42
\]
\[
46x \ge -35
\]

### Step 5: Divide both sides by 46 to solve for \(x\).
\[
x \ge \frac{-35}{46}
\]

### Step 6: Simplify the fraction if possible.
\[
x \ge \frac{-35}{46}
\]
The fraction \(\frac{-35}{46}\) cannot be simplified further, so the solution is:
\[
x \ge \frac{-35}{46}
\]

This is the final answer.

Would you like further details on this solution, or do you have any questions?

### Related Questions:
1. How do you solve inequalities involving distribution?
2. What happens if you multiply or divide both sides of an inequality by a negative number?
3. How can you graph the solution of an inequality on a number line?
4. What are the rules for combining like terms in algebraic expressions?
5. How does the solution change if the inequality sign is reversed?

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

Learn how to solve the inequality 8(6x + 5) + 2 ≥ 2x + 7 with a detailed step-by-step solution. This problem covers key algebraic concepts, including inequalities, distribution, and combining like terms. Suitable for students in Grades 9-12.

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