Solve the inequality.
2 x minus 2 less than 2 left parenthesis x minus 3 right parenthesis
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
Solve the inequality.
9xless than or equals9 left parenthesis x minus 6 right parenthesis
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is 
  
enter your response here.
​(Simplify your answer. Type your answer in interval​ notation.)
B.
The solution is the empty set.

Let's solve the inequalities step by step.

### First Inequality:
\[ 2x - 2 < 2(x - 3) \]

1. **Distribute** on the right side:
   \[
   2x - 2 < 2x - 6
   \]

2. **Subtract \(2x\)** from both sides to eliminate the \(2x\) terms:
   \[
   -2 < -6
   \]

This inequality is **not** correct as \(-2\) is not less than \(-6\). Therefore, this inequality has no solution.

### Second Inequality:
\[ 9x \leq 9(x - 6) \]

1. **Distribute** on the right side:
   \[
   9x \leq 9x - 54
   \]

2. **Subtract \(9x\)** from both sides to eliminate the \(9x\) terms:
   \[
   0 \leq -54
   \]

This inequality is also incorrect as \(0\) is not less than or equal to \(-54\). Therefore, the solution set is the **empty set**.

### Conclusion:

For both inequalities, the solution is the empty set.

### Follow-up Questions:

1. What happens when the inequality symbol is reversed?
2. Can you explain what interval notation is and how it relates to inequalities?
3. How would the solution change if the constants were positive instead?
4. What is the difference between a strict inequality and a non-strict inequality?
5. How can these inequalities be graphically represented on a number line?

### Tip:
Always check your results by substituting values into the original inequality to ensure correctness.

Solve the inequality. 2x - 2 < 2(x - 3) and 9x ≤ 9(x - 6).

Master solving algebraic inequalities with detailed examples, including 2x - 2 < 2(x - 3) and 9x ≤ 9(x - 6). This guide covers key concepts like the distributive property and properties of inequalities, ideal for Grades 8-10.

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