(x/4-1/x)/(1/2x+1/4)

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression (x/4 - 1/x) / (1/2x + 1/4)

This article will guide you through the steps of simplifying the complex rational expression: (x/4 - 1/x) / (1/2x + 1/4).

Step 1: Finding a Common Denominator for the Numerator

• Focus on the numerator: (x/4 - 1/x)
• Identify the least common denominator: The least common denominator for 4 and x is 4x.
• Rewrite each term with the common denominator:
  • (x/4) * (x/x) = x²/4x
  • (1/x) * (4/4) = 4/4x
• Substitute back into the original expression: (x²/4x - 4/4x) / (1/2x + 1/4)

Step 2: Finding a Common Denominator for the Denominator

• Focus on the denominator: (1/2x + 1/4)
• Identify the least common denominator: The least common denominator for 2x and 4 is 4x.
• Rewrite each term with the common denominator:
  • (1/2x) * (2/2) = 2/4x
  • (1/4) * (x/x) = x/4x
• Substitute back into the original expression: (x²/4x - 4/4x) / (2/4x + x/4x)

Step 3: Simplifying the Expression

• Combine the terms in the numerator and denominator: (x² - 4) / 4x / (2 + x) / 4x
• Simplify by dividing by the common denominator: (x² - 4) / (2 + x)

Step 4: Factoring the Numerator

• Factor the numerator: (x + 2)(x - 2) / (2 + x)
• Simplify by cancelling the common factor: x - 2

Final Result

The simplified form of the expression (x/4 - 1/x) / (1/2x + 1/4) is x - 2.

Important Note: This simplification assumes that x ≠ 0 and x ≠ -2. These values would make the original expression undefined.