https://file.aiquickdraw.com/mathbot/file/1730534151975p1xefzb2.jpg

The equation shown on the board is:

\[
16 \times \sqrt{\frac{3x + 1}{2x + 1}} = 8 - \sqrt{\frac{x}{3x + 1}}
\]

and on the right side, there's an expression:

\[
x = 8
\]

### Solution

Let's substitute \( x = 8 \) into the equation to see if both sides are equal, or whether we can simplify the equation further.

1. **Substitute \( x = 8 \):**
   
   For the left side:
   \[
   16 \times \sqrt{\frac{3(8) + 1}{2(8) + 1}} = 16 \times \sqrt{\frac{24 + 1}{16 + 1}} = 16 \times \sqrt{\frac{25}{17}}
   \]
   Simplify this further:
   \[
   = 16 \times \sqrt{\frac{25}{17}} = 16 \times \frac{5}{\sqrt{17}} = \frac{80}{\sqrt{17}}
   \]
   
2. **Right side substitution:**

   For the right side:
   \[
   8 - \sqrt{\frac{8}{3(8) + 1}} = 8 - \sqrt{\frac{8}{24 + 1}} = 8 - \sqrt{\frac{8}{25}} = 8 - \frac{\sqrt{8}}{5}
   \]
   We can further simplify \( \sqrt{8} \) as \( 2\sqrt{2} \):
   \[
   = 8 - \frac{2\sqrt{2}}{5}
   \]

Now we have both sides in terms of numerical values with radicals, which can be compared or approximated. Alternatively, we can test this algebraically or approximate both sides for equality.

Would you like a full numeric approximation or a further detailed explanation of each step?

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### Follow-Up Questions

1. What if we chose another value for \( x \) instead of \( 8 \)? Would the equation still hold?
2. Can we eliminate the square roots by squaring both sides?
3. What happens if we isolate \( \sqrt{\frac{3x + 1}{2x + 1}} \) on one side before substituting values?
4. How can we check if the equation is true for all values of \( x \), or only specific ones?
5. Is it possible to simplify \( \sqrt{\frac{3x + 1}{2x + 1}} \) and \( \sqrt{\frac{x}{3x + 1}} \) independently for easier comparison?

---

### Tip
To simplify equations with square roots, try squaring both sides carefully. This can sometimes help eliminate the radicals and make solving easier.

Solve the equation: 16 * sqrt((3x + 1) / (2x + 1)) = 8 - sqrt(x / (3x + 1)), and verify if x = 8 is a solution.

Explore the solution to the equation 16 * sqrt((3x + 1) / (2x + 1)) = 8 - sqrt(x / (3x + 1)) and verify if x = 8 satisfies it. The solution includes steps for handling radicals and simplifying square roots, suitable for high school students studying Algebra.

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