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The equation given is:

\[
\frac{5}{x-3} + \frac{2}{x+1} = 1
\]

We need to solve this equation for \(x\) and provide the answer correct to 3 significant figures.

### Steps:

1. **Eliminate the denominators** by multiplying both sides of the equation by \((x - 3)(x + 1)\), the least common denominator (LCD).

\[
(x - 3)(x + 1) \left( \frac{5}{x-3} + \frac{2}{x+1} \right) = (x - 3)(x + 1) \cdot 1
\]

2. **Simplify**:
   - The first term simplifies as: \((x - 3)(x + 1) \cdot \frac{5}{x-3} = 5(x + 1)\)
   - The second term simplifies as: \((x - 3)(x + 1) \cdot \frac{2}{x+1} = 2(x - 3)\)
   - The right-hand side simplifies as: \((x - 3)(x + 1)\)

This gives:

\[
5(x + 1) + 2(x - 3) = (x - 3)(x + 1)
\]

3. **Expand both sides**:
   - Expand the left-hand side:
     \[
     5(x + 1) = 5x + 5
     \]
     \[
     2(x - 3) = 2x - 6
     \]
     Combining these:
     \[
     5x + 5 + 2x - 6 = 7x - 1
     \]

   - Expand the right-hand side:
     \[
     (x - 3)(x + 1) = x^2 + x - 3x - 3 = x^2 - 2x - 3
     \]

So, the equation becomes:

\[
7x - 1 = x^2 - 2x - 3
\]

4. **Rearrange** the equation into standard quadratic form:
\[
0 = x^2 - 2x - 3 - 7x + 1
\]
\[
0 = x^2 - 9x - 2
\]

5. **Solve the quadratic equation** \(x^2 - 9x - 2 = 0\) using the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-2)}}{2(1)}
\]
\[
x = \frac{9 \pm \sqrt{81 + 8}}{2}
\]
\[
x = \frac{9 \pm \sqrt{89}}{2}
\]
\[
x = \frac{9 \pm 9.434}{2}
\]

Thus, the two solutions are:

\[
x_1 = \frac{9 + 9.434}{2} = \frac{18.434}{2} = 9.217
\]
\[
x_2 = \frac{9 - 9.434}{2} = \frac{-0.434}{2} = -0.217
\]

### Final Answer:
- \(x_1 = 9.22\) (to 3 significant figures)
- \(x_2 = -0.217\) (to 3 significant figures)

Let me know if you need further clarification or details!

---

**Related questions:**
1. How do you find the common denominator for rational equations?
2. What are the steps to expanding binomials like \((x - 3)(x + 1)\)?
3. How does the quadratic formula work, and when is it used?
4. What are the possible interpretations of the discriminant in a quadratic equation?
5. How do you verify solutions for a rational equation?

**Tip:** Always double-check the domain of rational functions to ensure the solutions do not make any denominators zero.

Solve the equation 5/(x-3) + 2/(x+1) = 1. Give your answer correct to 3 sf.

Learn how to solve the rational equation 5/(x-3) + 2/(x+1) = 1 using the quadratic formula. Detailed step-by-step explanation provided for students in Grades 9-11, including how to eliminate the denominators and apply the quadratic formula.

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