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2x−1−53x−2\frac{2}{x}-1 - \frac{5}{3x}-2

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Full solution

Q. 2x−1−53x−2\frac{2}{x}-1 - \frac{5}{3x}-2
  1. Find Common Denominator: Find a common denominator for the two fractions.\newlineThe common denominator is (x−1)(3x−2)(x-1)(3x-2).
  2. Rewrite with Common Denominator: Rewrite each fraction with the common denominator.\newline2x−1⋅3x−23x−2−53x−2⋅x−1x−1\frac{2}{x-1} \cdot \frac{3x-2}{3x-2} - \frac{5}{3x-2} \cdot \frac{x-1}{x-1}
  3. Multiply Numerators: Multiply the numerators by the appropriate expressions.\newline(2×(3x−2))/((x−1)(3x−2))−(5×(x−1))/((x−1)(3x−2))(2 \times (3x-2)) / ((x-1)(3x-2)) - (5 \times (x-1)) / ((x-1)(3x-2))
  4. Distribute Numerators: Distribute the numerators.\newline(6x−4)/((x−1)(3x−2))−(5x−5)/((x−1)(3x−2))(6x - 4) / ((x-1)(3x-2)) - (5x - 5) / ((x-1)(3x-2))
  5. Combine Numerators: Combine the numerators over the common denominator. (6x−4)−(5x−5)(x−1)(3x−2)\frac{(6x - 4) - (5x - 5)}{(x-1)(3x-2)}
  6. Simplify Numerator: Simplify the numerator. (6x−4−5x+5)/((x−1)(3x−2))(6x - 4 - 5x + 5) / ((x-1)(3x-2))
  7. Combine Like Terms: Combine like terms in the numerator. \newline(x+1)/((x−1)(3x−2))(x + 1) / ((x-1)(3x-2))
  8. Check for Simplification: Check for any possible simplification.\newlineNo further simplification is possible.

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