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You've provided an inequality, \(-47 < -4r - 7 \leq -35\), and a number line indicating potential solutions. Let's solve the inequality step-by-step:

1. Solve the first part of the inequality, \(-47 < -4r - 7\):
   \[
   -47 + 7 < -4r \quad \Rightarrow \quad -40 < -4r \quad \Rightarrow \quad r > 10
   \]
   (Dividing by \(-4\) reverses the inequality)

2. Solve the second part of the inequality, \(-4r - 7 \leq -35\):
   \[
   -4r - 7 + 7 \leq -35 + 7 \quad \Rightarrow \quad -4r \leq -28 \quad \Rightarrow \quad r \geq 7
   \]
   (Again, dividing by \(-4\) reverses the inequality)

Combining both parts, we have:
\[ 
10 < r \leq 12
\]
This range, \(r\) values between 10 and 12 inclusive of 12, is indicated by an open circle at 10 (not including 10) and a filled circle at 12 (including 12), matching the line segment shown on your number line.

Would you like more details on any step or have another question?

Here are some related questions you might find interesting:
1. How would you graph this inequality on a number line?
2. What happens if you multiply or divide an inequality by a negative number?
3. Can you explain why the inequality sign reverses when you divide by a negative number?
4. What would be the solution if the inequalities were reversed?
5. How can this type of inequality be applied in real-life situations?

**Tip:** Always check your solution by substituting a value from the solution set back into the original inequality to ensure it holds true.

-47 < -4r - 7 ≤ -35, with a number line solution

Learn how to solve the inequality -47 < -4r - 7 ≤ -35 and represent the solution on a number line. This detailed step-by-step solution covers key algebraic concepts such as inequalities and reversing inequality signs when dividing by negative numbers. Suitable for students in Grades 7-9.

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