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To solve for \( x \) in this right triangle, we can use trigonometric ratios.

### Step-by-Step Solution

1. **Identify the sides relative to the angle:**
   - We have a right triangle with a given angle of \( 31^\circ \).
   - The side opposite this angle (31°) is labeled \( x \).
   - The hypotenuse of the triangle is given as \( 15 \).

2. **Choose the trigonometric function:**
   - Since we know the hypotenuse and are looking for the side opposite the given angle, we can use the **sine function**, which is defined as:
     \[
     \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
     \]
   - Here, \( \theta = 31^\circ \), the opposite side is \( x \), and the hypotenuse is \( 15 \).

3. **Set up the equation:**
   \[
   \sin(31^\circ) = \frac{x}{15}
   \]

4. **Solve for \( x \):**
   - Rearrange the equation to isolate \( x \):
     \[
     x = 15 \cdot \sin(31^\circ)
     \]

5. **Calculate \( x \):**
   - Using a calculator to find \( \sin(31^\circ) \approx 0.5150 \):
     \[
     x \approx 15 \cdot 0.5150 = 7.725
     \]

6. **Round the answer to the nearest tenth:**
   \[
   x \approx 7.7
   \]

### Answer
\[
x = 7.7
\]

Would you like more details, or do you have any questions?

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Here are some related questions for further practice:

1. How would you solve for the hypotenuse if the opposite side and angle were given?
2. What if the adjacent side and the angle were given—how would you solve for \( x \)?
3. Can you use cosine or tangent to solve this problem instead? Why or why not?
4. How does the value of \( x \) change if the angle increases or decreases?
5. What other trigonometric ratios are useful in solving right triangle problems?

**Tip:** Always double-check which side (opposite, adjacent, or hypotenuse) you’re working with before choosing a trigonometric function.

Solve for x in the triangle. Round your answer to the nearest tenth.

Learn how to solve for the unknown side x in a right triangle using trigonometric ratios. This problem includes calculating sine for an angle, setting up the equation, and rounding to the nearest tenth. Suitable for students in Grades 9-10.

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