Solve for $x$ . $\newline$ $\dfrac{-2x-5}{5x-3} + \dfrac{6x-16}{6-10x} = \dfrac{-1}{2}$

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Grade 8

Understanding negative exponents

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Q. Solve for

x

\newline

\dfrac{-2x-5}{5x-3} + \dfrac{6x-16}{6-10x} = \dfrac{-1}{2}

Identify Common Denominator: Identify the common denominator for the fractions on the left side of the equation. $\newline$ The denominators are $(5x-3)$ and $(6-10x)$ . To combine these fractions, we need a common denominator, which will be the product of the two denominators: $(5x-3)(6-10x)$ .
Rewrite with Common Denominator: Rewrite each fraction with the common denominator. $\newline$ $\dfrac{-2x-5}{5x-3} \cdot \dfrac{6-10x}{6-10x} + \dfrac{6x-16}{6-10x} \cdot \dfrac{5x-3}{5x-3} = \dfrac{-1}{2}$
Distribute and Combine Fractions: Distribute the numerators and combine the fractions. $\newline$ $\dfrac{(-2x-5)(6-10x) + (6x-16)(5x-3)}{(5x-3)(6-10x)} = \dfrac{-1}{2}$
Expand Numerators: Expand the numerators. $\newline$ $\dfrac{(-2x)(6) + (-2x)(-10x) + (-5)(6) + (-5)(-10x) + (6x)(5x) + (6x)(-3) + (-16)(5x) + (-16)(-3)}{(5x-3)(6-10x)} = \dfrac{-1}{2}$
Simplify Numerator: Simplify the numerator by combining like terms. $\newline$ $\dfrac{-12x + 20x^2 - 30 + 50x + 30x^2 - 18x - 80x + 48}{(5x-3)(6-10x)} = \dfrac{-1}{2}$
Combine Like Terms: Combine like terms in the numerator. $\newline$ $\dfrac{20x^2 + 30x^2 - 12x + 50x - 18x - 80x - 30 + 48}{(5x-3)(6-10x)} = \dfrac{-1}{2}$ $\newline$ $\dfrac{50x^2 - 60x + 18}{(5x-3)(6-10x)} = \dfrac{-1}{2}$
Eliminate Fractions: Multiply both sides of the equation by the common denominator to eliminate the fractions. $\newline$ $(5x-3)(6-10x) \cdot \dfrac{50x^2 - 60x + 18}{(5x-3)(6-10x)} = \dfrac{-1}{2} \cdot (5x-3)(6-10x)$
Simplify Equations: Simplify both sides of the equation. $\newline$ $50x^2 - 60x + 18 = \dfrac{-1}{2} \cdot (5x-3)(6-10x)$
Distribute Right Side: Distribute the right side of the equation. $\newline$ $50x^2 - 60x + 18 = \dfrac{-1}{2} \cdot (30x - 50x^2 - 18x + 30)$
Combine Like Terms: Simplify the right side by combining like terms and multiplying by $-1$ / $2$ . $\newline$ $50x^2 - 60x + 18 = -15x + 25x^2 + 9x - 15$
Set Equation to Zero: Set the equation to zero by moving all terms to one side. $\newline$ $50x^2 - 60x + 18 - 25x^2 + 15x - 9x + 15 = 0$
Factor Quadratic Equation: Combine like terms. $\newline$ $25x^2 - 54x + 33 = 0$
Set Factors Equal to Zero: Factor the quadratic equation. $\newline$ $(5x - 3)(5x - 11) = 0$
Set Factors Equal to Zero: Factor the quadratic equation. $\newline$ $(5x - 3)(5x - 11) = 0$ Set each factor equal to zero and solve for x. $\newline$ $5x - 3 = 0$ or $5x - 11 = 0$ $\newline$ $5x = 3$ or $5x = 11$ $\newline$ $x = \dfrac{3}{5}$ or $x = \dfrac{11}{5}$