https://file.aiquickdraw.com/mathbot/file/1735929938222w8krjw3e.jpg

Fill in the blanks correctly, do not make your own solution

Here’s the solution based on the image provided. I will carefully fill in the blanks without introducing my own solution.

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### Step 19
- \( m\angle K < m\angle N \)
- **Substitute:** \( 3x + 18 < 90 \)
- **Subtract 18 from each side:** \( 3x < 72 \)
- **Divide each side by 3:** \( x < 24 \)

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### Step 20
- \( m\angle K > 0 \)
- **Substitute:** \( 3x + 18 > 0 \)
- **Subtract 18 from each side:** \( 3x > -18 \)
- **Divide each side by 3:** \( x > -6 \)

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### Step 21
The compound inequality becomes:
\[
-6 < x < 24
\]

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If you need further clarifications or have questions, let me know! 😊

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### 5 Related Questions:
1. What is the Converse of the Hinge Theorem, and how is it used in this problem?
2. Why does \( m\angle K \) need to be greater than 0?
3. How would the inequality change if the angle \( m\angle N \) was larger than 90°?
4. What are the implications if \( x \) falls outside the range \(-6 < x < 24\)?
5. How can this compound inequality be visualized on a number line?

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### Tip:
Always verify your solutions by substituting values within the range back into the original equations to ensure they satisfy the problem's conditions!

Learn how to find the range of possible values for x in a triangle using the Converse of the Hinge Theorem. This step-by-step solution involves solving inequalities for angle measures and understanding triangle properties, suitable for Grade 10 students.

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