But with inequalities don't you flip the sign (e.g <) every time you multiply or divide? that what I did in image 2

The rule for flipping the inequality sign only applies when you **multiply or divide by a negative number**. Let's break this down:

### Inequality Rule:
- When you multiply or divide all parts of an inequality by a **positive number**, the inequality sign stays the same.
- When you multiply or divide all parts of an inequality by a **negative number**, you must **flip** the inequality sign.

### Applying the Rule to Your Example:
For both images, the original inequality is:
\[
-3 \leq \frac{x}{2} \leq 5
\]
You are multiplying all parts of the inequality by 2, which is a **positive number**. Therefore, the inequality signs remain the same:

\[
-3 \times 2 \leq \frac{x}{2} \times 2 \leq 5 \times 2
\]

Simplifying, we get:
\[
-6 \leq x \leq 10
\]

### Image 2 Issue:
In Image 2, you flipped the inequality signs, but since you multiplied by a positive number (2), this was incorrect. The correct solution does not require flipping the signs.

### Correct Understanding:
- **Do not flip the inequality sign** when multiplying or dividing by a positive number.
- **Flip the inequality sign** only when multiplying or dividing by a negative number.

Would you like to explore more examples or have any further questions?

Here are some related questions you might find useful:
1. What happens if you multiply an inequality by a negative number?
2. Can you provide an example of solving an inequality where you need to flip the sign?
3. What is the effect of adding or subtracting a number on both sides of an inequality?
4. How do you graph inequalities with strict inequality symbols (< or >)?
5. What is the meaning of an "inclusive" inequality, like \( \leq \) or \( \geq \)?
6. How do compound inequalities work?
7. How do you solve inequalities that involve fractions?
8. Can you solve an inequality with a quadratic expression?

**Tip:** Always remember to double-check the sign of the number you are multiplying or dividing by when solving inequalities to determine if you need to flip the inequality sign.

Learn about inequalities and the sign flipping rule when multiplying or dividing by positive and negative numbers. This detailed explanation uses examples to clarify common misconceptions and ensures correct application of inequality signs.

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