2x + 2 > 8x + 6 

a. Make a prediction for the sign of x by comparing the expressions on either side of the inequality sign. 

Will it include positive or negative values? Explain. 

To predict the sign of \( x \), let's compare the two expressions on either side of the inequality: 

- The left side: \( 2x + 2 \)
- The right side: \( 8x + 6 \)

### Step 1: Analyze the Coefficients of \( x \)
- On the left side, the coefficient of \( x \) is \( 2 \).
- On the right side, the coefficient of \( x \) is \( 8 \).

Since \( 8x \) on the right side grows faster than \( 2x \) on the left side, the right side will tend to become larger as \( x \) increases.

### Step 2: Make a Prediction
For the inequality \( 2x + 2 > 8x + 6 \) to hold true, \( 2x + 2 \) must be greater than \( 8x + 6 \). However, since \( 8x \) grows faster than \( 2x \), this inequality is more likely to be true when \( x \) is negative or very small. As \( x \) becomes positive and larger, \( 8x + 6 \) will surpass \( 2x + 2 \), making the inequality false.

### Conclusion
The inequality will likely include **negative values** of \( x \) because the left side needs to stay greater than the right side, which is more difficult as \( x \) increases positively.

Would you like to see the detailed solution of the inequality or have any further questions?

Here are some related questions you might find helpful:
1. How do you solve linear inequalities step by step?
2. What happens when you graph the solution of an inequality?
3. How does changing the inequality sign affect the solution set?
4. What is the difference between strict inequalities and non-strict inequalities?
5. How can you test potential solutions to verify an inequality?

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

Understand how to predict the sign of x in linear inequalities with an example problem: 2x + 2 > 8x + 6. Explore step-by-step analysis, coefficients comparison, and when the inequality is likely true or false based on x values. Suitable for Grades 9-12.

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