Let's solve the inequality \( 2x + 2 > 8x + 6 \) step by step.

### Step 1: Move the \( x \)-terms to one side
Subtract \( 2x \) from both sides to get the \( x \)-terms on the right side:

\[
2x + 2 - 2x > 8x + 6 - 2x
\]

This simplifies to:

\[
2 > 6x + 6
\]

### Step 2: Move the constant terms to the other side
Subtract 6 from both sides to isolate the \( x \)-term:

\[
2 - 6 > 6x
\]

This simplifies to:

\[
-4 > 6x
\]

### Step 3: Solve for \( x \)
To isolate \( x \), divide both sides by 6:

\[
\frac{-4}{6} > x
\]

Simplifying the fraction:

\[
-\frac{2}{3} > x
\]

### Final Answer:
\[
x < -\frac{2}{3}
\]

So, the solution to the inequality is \( x < -\frac{2}{3} \). This means \( x \) must be any value less than \( -\frac{2}{3} \).

Would you like further details on how this solution relates to the initial prediction, or do you have any other questions?

Here are some related questions:
1. How do you graph the solution of \( x < -\frac{2}{3} \) on a number line?
2. What happens if you multiply or divide both sides of an inequality by a negative number?
3. How do you check if a specific value satisfies an inequality?
4. Can you solve compound inequalities?
5. What is the difference between solving equations and inequalities?

**Tip:** When solving inequalities, remember that if you multiply or divide by a negative number, the inequality sign must be reversed.

Learn how to solve the inequality 2x + 2 > 8x + 6 with a detailed step-by-step solution. This problem involves solving linear inequalities and manipulating algebraic expressions. Suitable for high school students.

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