Solve the inequality and graph the solution on the number line. \( 3 x+6<2 x+8 \) or \( 2 x+4 \leq 4 x-6 \)

To solve the inequality and graph the solution on the number line, we'll tackle each part of the compound inequality separately and then combine the results. ### Given Inequality: \[ 3x + 6 < 2x + 8 \quad \text{or} \quad 2x + 4 \leq 4x - 6 \] ### Solving Each Inequality: 1. **First Inequality:** \[ 3x + 6 < 2x + 8 \] - **Subtract \(2x\) from both sides:** \[ 3x - 2x + 6 < 8 \] \[ x + 6 < 8 \] - **Subtract 6 from both sides:** \[ x < 2 \] **Solution:** \( x < 2 \) 2. **Second Inequality:** \[ 2x + 4 \leq 4x - 6 \] - **Subtract \(2x\) from both sides:** \[ 4 \leq 2x - 6 \] - **Add 6 to both sides:** \[ 10 \leq 2x \] - **Divide both sides by 2:** \[ 5 \leq x \] \[ x \geq 5 \] **Solution:** \( x \geq 5 \) ### Combining the Solutions: The original compound inequality uses "or," meaning the solution is the union of both individual solutions. \[ x < 2 \quad \text{or} \quad x \geq 5 \] ### Graphing on the Number Line: 1. **For \( x < 2 \):** - Draw an **open circle** at \( x = 2 \) (since 2 is not included). - Shade the line to the **left** of 2. 2. **For \( x \geq 5 \):** - Draw a **closed circle** at \( x = 5 \) (since 5 is included). - Shade the line to the **right** of 5. ### Number Line Representation: ``` <---|-----------|===========> 2 5 ``` - **Open circle** at 2 (indicated by the parenthesis in the interval). - **Closed circle** at 5 (indicated by the square bracket in the interval). - Shading extends to the left of 2 and to the right of 5. ### Final Solution: \[ x < 2 \quad \text{or} \quad x \geq 5 \] ### Interval Notation: \[ (-\infty, 2) \cup [5, \infty) \]