Solve for the following inequalities: 3.1. Given: \( 4 \leq 3 x+1 \leq 7 \); where \( x \in R \) 3.1.1 Solve for \( x \) 3.1.2 Represent the answer to 4.1 in a number line.

To solve the compound inequality \( 4 \leq 3x + 1 \leq 7 \), we can break it down into two parts. First, let's solve the left part of the inequality: 1. \( 4 \leq 3x + 1 \) - Subtract 1 from both sides: \( 4 - 1 \leq 3x \) - This simplifies to \( 3 \leq 3x \) - Divide both sides by 3: \( 1 \leq x \) or \( x \geq 1 \) Next, solve the right part: 2. \( 3x + 1 \leq 7 \) - Subtract 1 from both sides: \( 3x \leq 7 - 1 \) - This simplifies to \( 3x \leq 6 \) - Divide both sides by 3: \( x \leq 2 \) Combining both parts, we get: \[ 1 \leq x \leq 2 \] This means \( x \) can take any value from 1 to 2, inclusive. Now, let's represent this on a number line: - Draw a horizontal line. - Mark the points 1 and 2 on the line. - Use a solid dot at 1 and 2 to indicate that they are included in the solution. - Shade the area between 1 and 2 to show that all values in this interval are solutions: it looks like this on a number line: \[ \text{--- •---|---•---} \] \[ \text{ 1 2} \] In conclusion, the solution to the inequality and its representation on a number line show that \( x \) is inclusive between 1 and 2.