The question asks to solve the three-part inequality analytically and support the answer graphically:

$1 \leq 2x + 1 \leq 5$

Let's solve this step-by-step:

Step 1: Break the Three-Part Inequality into Two Parts

1. Solve the left inequality: $1 \leq 2x + 1$
2. Solve the right inequality: $2x + 1 \leq 5$

Step 2: Solve Each Inequality Separately

Solving the Left Inequality: $1 \leq 2x + 1$

1. Subtract 1 from both sides:
   $1 - 1 \leq 2x + 1 - 1$ $0 \leq 2x$
2. Divide both sides by 2:
   $\frac{0}{2} \leq \frac{2x}{2}$ $0 \leq x$ Or simply:
   $x \geq 0$

Solving the Right Inequality: $2x + 1 \leq 5$

1. Subtract 1 from both sides:
   $2x + 1 - 1 \leq 5 - 1$ $2x \leq 4$
2. Divide both sides by 2:
   $\frac{2x}{2} \leq \frac{4}{2}$ $x \leq 2$

Step 3: Combine the Solutions

From both inequalities, we have:

$0 \leq x \leq 2$

Step 4: Write the Solution in Interval Notation

The solution set in interval notation is:

$[0, 2]$

Conclusion

The solution set for the three-part inequality $1 \leq 2x + 1 \leq 5$ is $[0, 2]$.

If you'd like further details or have any questions, feel free to ask!

Related Questions:

1. How do you graph a solution set on a number line?
2. What are the steps to solve compound inequalities?
3. How do you represent inequalities involving "less than" or "greater than" in interval notation?
4. Can you explain how to solve inequalities with absolute values?
5. How do you solve a system of linear inequalities?

Tip: When working with inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!