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:
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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.
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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.
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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.
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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: \( x < 8 \)