https://file.aiquickdraw.com/mathbot/file/1725740060522k1ols23w.jpg

Let's solve the inequality step by step:

We are given:
\[
1 \leq 2x + 5 < 8
\]

### Step 1: Break the compound inequality into two parts.

1. Solve the left side:
\[
1 \leq 2x + 5
\]
Subtract 5 from both sides:
\[
1 - 5 \leq 2x \quad \Rightarrow \quad -4 \leq 2x
\]
Now divide by 2:
\[
-2 \leq x
\]

2. Solve the right side:
\[
2x + 5 < 8
\]
Subtract 5 from both sides:
\[
2x < 3
\]
Now divide by 2:
\[
x < \frac{3}{2}
\]

### Step 2: Combine the two parts.

So, the solution as an inequality is:
\[
-2 \leq x < \frac{3}{2}
\]

### Step 3: Express in interval notation.

The interval notation for the solution is:
\[
[-2, \frac{3}{2})
\]
This means that \(x\) can take any value from \(-2\) to \(1.5\) (exclusive of \(1.5\)).

### Step 4: Plot on a number line.

- There will be a solid dot at \(x = -2\) because \(-2\) is included in the solution.
- There will be an open dot at \(x = \frac{3}{2}\) (or 1.5) because it is not included in the solution.
- The region between \(-2\) and \(1.5\) will be shaded.

---

Would you like a more detailed breakdown, or do you have any further questions? Here are some related questions to deepen your understanding:

1. How do you solve compound inequalities with different comparison signs (like \( \leq \) and \( < \))?
2. What happens if you multiply or divide an inequality by a negative number?
3. Can compound inequalities be written in different forms?
4. How does solving inequalities differ from solving equations?
5. What are some real-world examples where inequalities are used?

**Tip:** When dividing or multiplying both sides of an inequality by a negative number, always remember to reverse the inequality sign!

Solve the inequality: 1 <= 2x + 5 < 8. Provide the answer as an inequality, in interval notation, and on a number line.

Learn how to solve the compound inequality 1 <= 2x + 5 < 8 using algebraic methods. The solution is provided as an inequality, in interval notation, and visualized on a number line. Suitable for students in Grades 8-10.

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