https://file.aiquickdraw.com/mathbot/file/1725645004487yua78swd.jpg

The inequality given in the image is:

\[
-47 < -4r - 7 \leq -35
\]

To solve this, we'll handle both parts of the compound inequality step by step.

### Step 1: Solve the left inequality
We start with the left part:

\[
-47 < -4r - 7
\]

1. Add 7 to both sides:
   \[
   -47 + 7 < -4r
   \]
   \[
   -40 < -4r
   \]

2. Divide by -4, and remember to reverse the inequality when dividing by a negative number:
   \[
   \frac{-40}{-4} > \frac{-4r}{-4}
   \]
   \[
   10 > r
   \]
   This simplifies to:
   \[
   r < 10
   \]

### Step 2: Solve the right inequality
Now solve the right part:

\[
-4r - 7 \leq -35
\]

1. Add 7 to both sides:
   \[
   -4r \leq -35 + 7
   \]
   \[
   -4r \leq -28
   \]

2. Divide by -4, again reversing the inequality:
   \[
   \frac{-4r}{-4} \geq \frac{-28}{-4}
   \]
   \[
   r \geq 7
   \]

### Final solution:
Combining the two parts, the solution is:

\[
7 \leq r < 10
\]

This means that \( r \) is between 7 and 10, including 7 but not 10. The number line in the image also confirms this, as the interval from 7 to 10 is shaded, with a closed circle at 7 and an open circle at 10.

Would you like any further details on this problem?

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Here are five related questions:
1. How do you solve compound inequalities with variables on both sides?
2. What happens when you multiply or divide inequalities by negative numbers?
3. How do you graph solutions to inequalities on a number line?
4. What is the difference between strict inequalities and inclusive inequalities?
5. How can you represent compound inequalities using set notation?

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

-47 < -4r - 7 ≤ -35 (based on the image provided by the user)

Learn how to solve the compound inequality -47 < -4r - 7 ≤ -35 with a detailed step-by-step solution. This problem covers key algebraic concepts, including solving inequalities and understanding compound inequalities. Suitable for students in Grades 7-9.

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