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Solve for xx.\newline−2x−55x−3+6x−166−10x=−12\dfrac{-2x-5}{5x-3} + \dfrac{6x-16}{6-10x} = \dfrac{-1}{2}

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Q. Solve for xx.\newline−2x−55x−3+6x−166−10x=−12\dfrac{-2x-5}{5x-3} + \dfrac{6x-16}{6-10x} = \dfrac{-1}{2}
  1. Identify Common Denominator: Identify the common denominator for the fractions on the left side of the equation.\newlineThe denominators are (5x−3)(5x-3) and (6−10x)(6-10x). To combine these fractions, we need a common denominator, which will be the product of the two denominators: (5x−3)(6−10x)(5x-3)(6-10x).
  2. Rewrite with Common Denominator: Rewrite each fraction with the common denominator.\newline−2x−55x−3⋅6−10x6−10x+6x−166−10x⋅5x−35x−3=−12\dfrac{-2x-5}{5x-3} \cdot \dfrac{6-10x}{6-10x} + \dfrac{6x-16}{6-10x} \cdot \dfrac{5x-3}{5x-3} = \dfrac{-1}{2}
  3. Distribute and Combine Fractions: Distribute the numerators and combine the fractions.\newline(−2x−5)(6−10x)+(6x−16)(5x−3)(5x−3)(6−10x)=−12\dfrac{(-2x-5)(6-10x) + (6x-16)(5x-3)}{(5x-3)(6-10x)} = \dfrac{-1}{2}
  4. Expand Numerators: Expand the numerators.\newline(−2x)(6)+(−2x)(−10x)+(−5)(6)+(−5)(−10x)+(6x)(5x)+(6x)(−3)+(−16)(5x)+(−16)(−3)(5x−3)(6−10x)=−12\dfrac{(-2x)(6) + (-2x)(-10x) + (-5)(6) + (-5)(-10x) + (6x)(5x) + (6x)(-3) + (-16)(5x) + (-16)(-3)}{(5x-3)(6-10x)} = \dfrac{-1}{2}
  5. Simplify Numerator: Simplify the numerator by combining like terms.\newline−12x+20x2−30+50x+30x2−18x−80x+48(5x−3)(6−10x)=−12\dfrac{-12x + 20x^2 - 30 + 50x + 30x^2 - 18x - 80x + 48}{(5x-3)(6-10x)} = \dfrac{-1}{2}
  6. Combine Like Terms: Combine like terms in the numerator.\newline20x2+30x2−12x+50x−18x−80x−30+48(5x−3)(6−10x)=−12\dfrac{20x^2 + 30x^2 - 12x + 50x - 18x - 80x - 30 + 48}{(5x-3)(6-10x)} = \dfrac{-1}{2}\newline50x2−60x+18(5x−3)(6−10x)=−12\dfrac{50x^2 - 60x + 18}{(5x-3)(6-10x)} = \dfrac{-1}{2}
  7. Eliminate Fractions: Multiply both sides of the equation by the common denominator to eliminate the fractions.\newline(5x−3)(6−10x)⋅50x2−60x+18(5x−3)(6−10x)=−12⋅(5x−3)(6−10x)(5x-3)(6-10x) \cdot \dfrac{50x^2 - 60x + 18}{(5x-3)(6-10x)} = \dfrac{-1}{2} \cdot (5x-3)(6-10x)
  8. Simplify Equations: Simplify both sides of the equation.\newline50x2−60x+18=−12⋅(5x−3)(6−10x)50x^2 - 60x + 18 = \dfrac{-1}{2} \cdot (5x-3)(6-10x)
  9. Distribute Right Side: Distribute the right side of the equation.\newline50x2−60x+18=−12⋅(30x−50x2−18x+30)50x^2 - 60x + 18 = \dfrac{-1}{2} \cdot (30x - 50x^2 - 18x + 30)
  10. Combine Like Terms: Simplify the right side by combining like terms and multiplying by −1-1/22.\newline50x2−60x+18=−15x+25x2+9x−1550x^2 - 60x + 18 = -15x + 25x^2 + 9x - 15
  11. Set Equation to Zero: Set the equation to zero by moving all terms to one side.\newline50x2−60x+18−25x2+15x−9x+15=050x^2 - 60x + 18 - 25x^2 + 15x - 9x + 15 = 0
  12. Factor Quadratic Equation: Combine like terms.\newline25x2−54x+33=025x^2 - 54x + 33 = 0
  13. Set Factors Equal to Zero: Factor the quadratic equation.\newline(5x−3)(5x−11)=0(5x - 3)(5x - 11) = 0
  14. Set Factors Equal to Zero: Factor the quadratic equation.\newline(5x−3)(5x−11)=0(5x - 3)(5x - 11) = 0Set each factor equal to zero and solve for x.\newline5x−3=05x - 3 = 0 or 5x−11=05x - 11 = 0\newline5x=35x = 3 or 5x=115x = 11\newlinex=35x = \dfrac{3}{5} or x=115x = \dfrac{11}{5}

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