https://file.aiquickdraw.com/mathbot/file/1727476388466ouww9b6f.jpg

The problem in the image asks to solve the radical equation:

\[
\sqrt{x + 4} - \sqrt{x - 1} = 1
\]

### Steps to solve:

1. **Isolate one square root**:
   \[
   \sqrt{x + 4} = 1 + \sqrt{x - 1}
   \]

2. **Square both sides** to eliminate the square root on the left:
   \[
   (\sqrt{x + 4})^2 = (1 + \sqrt{x - 1})^2
   \]
   This simplifies to:
   \[
   x + 4 = 1^2 + 2(1)(\sqrt{x - 1}) + (\sqrt{x - 1})^2
   \]
   \[
   x + 4 = 1 + 2\sqrt{x - 1} + (x - 1)
   \]
   Simplify:
   \[
   x + 4 = x + 1 + 2\sqrt{x - 1}
   \]

3. **Isolate the remaining square root**:
   Subtract \(x + 1\) from both sides:
   \[
   4 - 1 = 2\sqrt{x - 1}
   \]
   \[
   3 = 2\sqrt{x - 1}
   \]

4. **Square both sides again**:
   \[
   \left(\frac{3}{2}\right)^2 = (\sqrt{x - 1})^2
   \]
   \[
   \frac{9}{4} = x - 1
   \]
   Add 1 to both sides:
   \[
   x = \frac{9}{4} + 1 = \frac{9}{4} + \frac{4}{4} = \frac{13}{4}
   \]

### Check the solution:
Substitute \(x = \frac{13}{4}\) into the original equation:

\[
\sqrt{\frac{13}{4} + 4} - \sqrt{\frac{13}{4} - 1} = 1
\]

Simplify:
\[
\sqrt{\frac{29}{4}} - \sqrt{\frac{9}{4}} = 1
\]
\[
\frac{\sqrt{29}}{2} - \frac{3}{2} = 1
\]

This does **not** hold true, so **\(x = \frac{13}{4}\) is not a solution**.

### Conclusion:
There are no valid solutions to this equation, so the solution set is:

\[
\boxed{\emptyset}
\]

Would you like further clarification on any step? 

Here are 5 related questions:
1. How do you solve a system with two radical equations?
2. What methods are used for isolating terms in equations?
3. How can extraneous solutions arise in radical equations?
4. Why is it necessary to check solutions in radical equations?
5. What are some common mistakes when squaring both sides of an equation?

**Tip:** Always check for extraneous solutions after solving equations involving radicals, as squaring both sides can introduce non-valid solutions.

Solve the radical equation: \(\sqrt{x + 4} - \sqrt{x - 1} = 1\)

This page walks you through solving the radical equation √(x+4) - √(x-1) = 1, using algebraic techniques to isolate terms, square both sides, and check for extraneous solutions. Learn how to solve step-by-step for Grades 10-12.

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