Question

Solving the inequality for x, which statement shows a good first step and which is the correct solution?

x−6<x4(2 points)
Responses

Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x


Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x

Divide both sides of the inequality by 4 to simplify: x4−32<x


Divide both sides of the inequality by 4 to simplify: x4−32<x

Multiply both sides by x to get the "x" terms on the same side of the inequality: x−6<4



Multiply both sides by x to get the "x" terms on the same side of the inequality: x−6<4

Divide both sides of the inequality by x to simplify: 1−6x<4
x<6

Answer 1

To solve the inequality \( x - 6 < \frac{x}{4} \), we need to manipulate the terms appropriately. Let's examine the provided options for the best first step and then find the correct solution.

### Analyzing the First Steps:

1. **Multiply both sides by 4 to clear the inequality of fractions:**
\[
4(x - 6) < x
\]
This can be expanded to:
\[
4x - 24 < x
\]
This is a valid first step because it eliminates the fraction.

2. **Divide both sides of the inequality by 4 to simplify:**
\[
\frac{x - 6}{4} < x
\]
This isn’t a great first step, as it keeps the fraction and could complicate things.

3. **Multiply both sides by x to get the "x" terms on the same side of the inequality:**
This step could lead to mistakes, especially if \( x \) is negative, as it changes the direction of the inequality.

4. **Divide both sides of the inequality by x to simplify:**
This step is also inappropriate without knowing the sign of \( x \).

### Conclusion for the First Step:
The best first step is:
- **Multiply both sides by 4 to clear the inequality of fractions: \( 4x - 24 < x \)**

### Solving the Inequality:
Now, taking the first step we found valid:
\[
4x - 24 < x
\]

Subtract \( x \) from both sides:
\[
4x - x - 24 < 0
\]
This simplifies to:
\[
3x - 24 < 0
\]

Next, add 24 to both sides:
\[
3x < 24
\]

Now, divide by 3:
\[
x < 8
\]

### Correct Solution:
Thus, the solution to the inequality is:
\[
x < 8
\]

### Summary:
- **Best First Step:** Multiply both sides by 4 to clear the fraction.
- **Correct Solution:** \( x < 8 \)