Given the following linear inequality: \( -6<2 x-4 \leq 2 \) 1.1.1 Solve the linear inequality. 1.1.2 Represent your solution on a number line.

### 1.1.1. Solving the Linear Inequality Given the inequality: \[ -6 < 2x - 4 \leq 2 \] **Step 1:** **Isolate the term with \( x \).** Add **4** to all parts of the inequality to eliminate the constant term on the left side: \[ -6 + 4 < 2x - 4 + 4 \leq 2 + 4 \] \[ -2 < 2x \leq 6 \] **Step 2:** **Solve for \( x \).** Divide all parts of the inequality by **2** to solve for \( x \): \[ \frac{-2}{2} < \frac{2x}{2} \leq \frac{6}{2} \] \[ -1 < x \leq 3 \] **Solution:** \[ x \in (-1, \, 3] \] This means \( x \) is greater than **-1** and less than or equal to **3**. ### 1.1.2. Representing the Solution on a Number Line To represent the solution \( -1 < x \leq 3 \) on a number line: 1. **Draw a horizontal line** and mark points **-1** and **3** on it. 2. **Open Circle at -1:** - Since \( x \) is **greater than** -1 but not equal to -1, use an **open circle** at **-1**. 3. **Closed Circle at 3:** - Since \( x \) is **less than or equal to** 3, use a **closed circle** at **3**. 4. **Shade the Region Between -1 and 3:** - Shade the line connecting the two points to indicate all values of \( x \) that satisfy the inequality. **Illustration:** ``` <-----|---------|=========|---------|-----> -2 -1 3 4 ( )=================● ``` - **( )** represents an open circle at **-1**. - **●** represents a closed circle at **3**. - **=** represents the shaded area between **-1** and **3**. **Summary:** - \( x \) can be any real number **greater than -1** and **up to 3**. - The number line visually displays all valid solutions within this range.