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The problem asks for the equivalent form of the inequality:

\[
2(x + 7) \geq 4x - 12
\]

Let's solve this step by step:

### Step 1: Expand the left-hand side
Distribute the 2 on the left-hand side:

\[
2(x + 7) = 2x + 14
\]

So the inequality becomes:

\[
2x + 14 \geq 4x - 12
\]

### Step 2: Move all terms involving \(x\) to one side
Subtract \(2x\) from both sides:

\[
14 \geq 2x - 12
\]

### Step 3: Move constant terms to the other side
Add 12 to both sides:

\[
14 + 12 \geq 2x
\]

\[
26 \geq 2x
\]

### Step 4: Solve for \(x\)
Divide both sides by 2:

\[
13 \geq x
\]

Or equivalently:

\[
x \leq 13
\]

### Final Answer:
The equivalent inequality is \(x \leq 13\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the inequality change if it was \(2(x + 7) > 4x - 12\)?
2. What if the inequality had been \(2(x - 7) \geq 4x - 12\)?
3. How would you solve the inequality if it involved fractions, like \( \frac{2}{3}(x + 7) \geq \frac{4}{5}x - 12\)?
4. What happens to the inequality sign if you multiply or divide by a negative number?
5. How can inequalities be represented graphically?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number in an inequality.

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Learn how to solve the inequality 2(x + 7) ≥ 4x - 12 step-by-step. This problem demonstrates key algebraic concepts like the distributive property and properties of inequalities. Suitable for students in Grades 8-10.

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