Question

Solve the rational equation 4/x−2+1=7/x−2 . Re-order the steps so that the given equation is at the top, the steps are ordered appropriately, and the solution is at the bottom. (1 point) Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.

Answer 1

To solve the rational equation \( \frac{4}{x-2} + 1 = \frac{7}{x-2} \), we'll follow these steps in the correct order:

1. **Rewrite the equation**: \( \frac{4}{x-2} + 1 = \frac{7}{x-2} \)

2. **Combine like terms**: Move \( \frac{7}{x-2} \) to the left side:
\( \frac{4}{x-2} + 1 - \frac{7}{x-2} = 0 \)

3. **Combine the fractions**: Since \( \frac{4}{x-2} - \frac{7}{x-2} = \frac{-3}{x-2} \), we rewrite it as:
\( \frac{-3}{x-2} + 1 = 0 \)

4. **Isolate the rational term**: Subtract 1 from both sides:
\( \frac{-3}{x-2} = -1 \)

5. **Multiply both sides by \( x-2 \)** (noting that \( x \neq 2 \)):
\( -3 = -1(x - 2) \)

6. **Distribute the right side**:
\( -3 = -x + 2 \)

7. **Rearranging the equation**: Adding \( x \) to both sides:
\( x - 3 = 2 \)

8. **Add 3 to both sides**:
\( x = 5 \)

9. **Final solution**:
The solution to the equation \( \frac{4}{x-2} + 1 = \frac{7}{x-2} \) is \( x = 5 \).

So the steps arranged appropriately to solve the equation are:

1. \( \frac{4}{x-2} + 1 = \frac{7}{x-2} \)
2. \( \frac{4}{x-2} + 1 - \frac{7}{x-2} = 0 \)
3. \( \frac{-3}{x-2} + 1 = 0 \)
4. \( \frac{-3}{x-2} = -1 \)
5. \( -3 = -1(x - 2) \)
6. \( -3 = -x + 2 \)
7. \( x - 3 = 2 \)
8. \( x = 5 \)

Final solution: \( x = 5 \)