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Let's solve the compound inequality \(-2 \leq 6 - 2x < 12\).

### Step-by-step Solution:

#### 1. Solve each part of the compound inequality separately.

The inequality consists of two parts:
- \( -2 \leq 6 - 2x \)
- \( 6 - 2x < 12 \)

---

**First part: \( -2 \leq 6 - 2x \)**

Subtract 6 from both sides:

\[
-2 - 6 \leq -2x \implies -8 \leq -2x
\]

Divide by \(-2\) (remember to flip the inequality sign when dividing by a negative number):

\[
\frac{-8}{-2} \geq x \implies 4 \geq x \implies x \leq 4
\]

---

**Second part: \( 6 - 2x < 12 \)**

Subtract 6 from both sides:

\[
6 - 6 - 2x < 12 - 6 \implies -2x < 6
\]

Divide by \(-2\) (again, flip the inequality sign):

\[
\frac{-2x}{-2} > \frac{6}{-2} \implies x > -3
\]

---

### 2. Combine the two inequalities.

We now have:
\[
x \leq 4 \quad \text{and} \quad x > -3
\]

The combined solution is:
\[
-3 < x \leq 4
\]

### 3. Write the solution in interval notation.

The interval notation for the solution set is:
\[
(-3, 4]
\]

### 4. Graph the solution set.

- A parenthesis \(()\) at \(-3\) indicates that \(-3\) is not included.
- A bracket \([]\) at \(4\) indicates that \(4\) is included.

The graph of the solution set on a number line would look like this:

\[
\text{Open circle at } -3, \text{ closed circle at } 4.
\]
- The line between them represents all values between \(-3\) and \(4\), with \(-3\) excluded and \(4\) included.

---

Would you like a detailed graph of this solution?

### Related Questions:
1. How do you solve compound inequalities involving fractions?
2. How would the solution change if the inequality was strict (i.e., without \(\leq\))?
3. What is the effect of multiplying or dividing by a negative number in inequalities?
4. How do you express the solution of a system of inequalities in set notation?
5. What is the difference between an open interval and a closed interval in inequalities?

### Tip:
Always remember to flip the inequality sign when multiplying or dividing by a negative number.

Solve and write interval notation for the solution set: -2 <= 6 - 2x < 12. Then graph the solution set.

Learn how to solve the compound inequality -2 <= 6 - 2x < 12 and write the solution in interval notation. This problem focuses on solving linear inequalities and understanding interval notation, suitable for students in Grades 7-9.

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