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Fully simplify to one fraction.\newline7x+6−xx2−x−42\frac{7}{x+6}-\frac{x}{x^{2}-x-42}

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Full solution

Q. Fully simplify to one fraction.\newline7x+6−xx2−x−42\frac{7}{x+6}-\frac{x}{x^{2}-x-42}
  1. Factor Denominator: Factor the denominator of the second fraction.\newlineThe denominator x2−x−42x^2 - x - 42 can be factored into (x−7)(x+6)(x - 7)(x + 6) because it is a quadratic expression.\newlineCalculation: x2−x−42=(x−7)(x+6)x^2 - x - 42 = (x - 7)(x + 6)
  2. Rewrite with Factored Denominator: Rewrite the expression with the factored denominator.\newlineNow that we have factored the denominator of the second fraction, we can rewrite the expression as:\newline(7)/(x+6)−(x)/((x−7)(x+6))(7)/(x+6) - (x)/((x - 7)(x + 6))
  3. Find Common Denominator: Find a common denominator for the two fractions.\newlineThe common denominator for the two fractions is x - \(7)(x + 66)\.
  4. Rewrite with Common Denominator: Rewrite both fractions with the common denominator.\newlineWe can now rewrite both fractions with the common denominator:\newline(7×(x−7))/((x−7)(x+6))−(x)/((x−7)(x+6))(7 \times (x - 7))/((x - 7)(x + 6)) - (x)/((x - 7)(x + 6))
  5. Combine Fractions: Combine the fractions over the common denominator.\newlineNow that both fractions have the same denominator, we can combine them into a single fraction:\newline(7∗(x−7)−x)/((x−7)(x+6))(7 * (x - 7) - x)/((x - 7)(x + 6))
  6. Expand and Simplify: Expand the numerator and simplify.\newlineExpand the numerator by distributing the 77 into (x−7)(x - 7) and then combine like terms:\newline7x−49−x(x−7)(x+6)\frac{7x - 49 - x}{(x - 7)(x + 6)}\newlineNow combine like terms in the numerator:\newline6x−49(x−7)(x+6)\frac{6x - 49}{(x - 7)(x + 6)}
  7. Check for Simplification: Check for any further simplification.\newlineThe numerator and the denominator have no common factors, so the expression cannot be simplified further.
  8. Write Final Answer: Write the final answer.\newlineThe fully simplified expression is:\newline(6x−49)/((x−7)(x+6))(6x - 49)/((x - 7)(x + 6))

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