https://file.aiquickdraw.com/mathbot/file/1725831586377aso8bcvp.jpg

The inequality in the image is:

\[
-7 \leq 8x + 2 < -1
\]

Let's solve it step by step.

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

1. \(-7 \leq 8x + 2\)
2. \(8x + 2 < -1\)

### Step 2: Solve each part.

#### Part 1: \(-7 \leq 8x + 2\)
- Subtract 2 from both sides:
  \[
  -7 - 2 \leq 8x
  \]
  \[
  -9 \leq 8x
  \]
- Divide both sides by 8:
  \[
  \frac{-9}{8} \leq x
  \]
  So, \(x \geq -\frac{9}{8}\).

#### Part 2: \(8x + 2 < -1\)
- Subtract 2 from both sides:
  \[
  8x < -1 - 2
  \]
  \[
  8x < -3
  \]
- Divide both sides by 8:
  \[
  x < \frac{-3}{8}
  \]
  So, \(x < -\frac{3}{8}\).

### Step 3: Combine the two parts.

The solution is:
\[
-\frac{9}{8} \leq x < -\frac{3}{8}
\]

### Step 4: Graph the solution.

On a number line, this is represented by a closed circle at \(x = -\frac{9}{8}\) (since \(-\frac{9}{8}\) is included) and an open circle at \(x = -\frac{3}{8}\) (since \(-\frac{3}{8}\) is not included), with a shaded region between them.

Would you like any further details on this solution or the graph?

Here are some related questions to explore:
1. How do we handle strict inequalities when graphing on a number line?
2. What happens if we divide both sides of an inequality by a negative number?
3. How do you interpret compound inequalities in real-life problems?
4. Can a solution to an inequality ever include more than one interval?
5. How can inequalities be useful in determining the solution sets for functions?

**Tip:** When solving inequalities, always remember that multiplying or dividing by a negative number flips the inequality sign!

Solve the compound inequality: -7 ≤ 8x + 2 < -1 and represent it on a number line.

Learn how to solve the compound inequality -7 ≤ 8x + 2 < -1 and graph the solution on a number line. This step-by-step guide covers algebraic techniques for solving inequalities and visual representation on a number line. Suitable for students in Grades 7-9.

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