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To solve for the length of side \( x \) in this triangle, we can use the Law of Sines, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

where \( a, b, c \) are the sides opposite the angles \( A, B, C \) respectively.

The angles given are:

- \( A = 32^\circ \) (opposite to side \( x \))
- \( B = 48^\circ \) (opposite to side \( 6 \))
- \( C = 180^\circ - 32^\circ - 48^\circ = 100^\circ \) (third angle of the triangle)

Using the Law of Sines:

\[
\frac{x}{\sin(32^\circ)} = \frac{6}{\sin(48^\circ)}
\]

Now, let's solve for \( x \):

\[
x = \frac{6 \cdot \sin(32^\circ)}{\sin(48^\circ)}
\]

Let's calculate the value of \( x \).The length of side \( x \) is approximately \( 4.28 \) units.

Would you like any additional details or have any other questions?

Here are five questions you might consider asking:
1. How do you use the Law of Sines to solve different types of triangles?
2. What is the Law of Cosines and how does it differ from the Law of Sines?
3. How can you find the area of a triangle given two sides and the included angle?
4. What are some common mistakes to avoid when applying trigonometric laws?
5. How can you verify the accuracy of your trigonometric calculations?

**Tip:** Always double-check the angles and sides you are given in a problem to ensure they are used correctly in trigonometric formulas.

Learn how to solve for the length of side x in a triangle using the Law of Sines with a detailed step-by-step solution. This problem covers key trigonometric concepts, including the Law of Sines formula. Suitable for high school students (Grades 10-12).

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