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

To solve the inequality \( -6 < 2x - 4 \leq 2 \) and represent its solution on a number line, follow these steps: ### Step 1: Solve the Inequality 1. **Start with the original inequality:** \[ -6 < 2x - 4 \leq 2 \] 2. **Add 4 to all parts of the inequality to isolate the term with \( x \):** \[ -6 + 4 < 2x - 4 + 4 \leq 2 + 4 \] \[ -2 < 2x \leq 6 \] 3. **Divide all parts by 2 to solve for \( x \):** \[ \frac{-2}{2} < \frac{2x}{2} \leq \frac{6}{2} \] \[ -1 < x \leq 3 \] ### Step 2: Represent the Solution on a Number Line The solution \( -1 < x \leq 3 \) can be visualized on a number line as follows: 1. **Draw a horizontal line and mark points for \( -1 \) and \( 3 \).** 2. **Use an open circle (◯) at \( -1 \) to indicate that \( -1 \) is **not** included in the solution.** 3. **Use a closed (filled) circle (●) at \( 3 \) to indicate that \( 3 \) **is** included in the solution.** 4. **Shade the region between \( -1 \) and \( 3 \) to show all real numbers \( x \) that satisfy the inequality.** Here's a visual representation: ``` <---|---------|=========● -1 3 ``` - **◯ at -1:** Not included - **● at 3:** Included - **Shaded line between -1 and 3:** All real numbers \( x \) such that \( -1 < x \leq 3 \) ### Summary - **Solution Set:** \( -1 < x \leq 3 \) - **Number Line Representation:** ``` <---|---------|=========● -1 3 ``` This indicates all real numbers greater than -1 and up to and including 3 satisfy the original inequality.