What value of x satisfies the equation xx - 2 - 4x + 3 = 10(x - 2)(x + 3)?

What value of x satisfies the equation xx - 2 - 4x + 3 = 10(x... | SAT Nonlinear equations in one variable and systems of equations in two variables Question, Answer Explained

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Math · Advanced MathHard

What value of $x$ satisfies the equation $\frac{x}{x - 2} - \frac{4}{x + 3} = \frac{10}{(x - 2)(x + 3)}$ ?

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Answer: -1

Multiply every term by $(x - 2)(x + 3)$ : $x(x + 3) - 4(x - 2) = 10$ . Expand: $x^2 + 3x - 4x + 8 = 10$ , so $x^2 - x - 2 = 0$ and $(x - 2)(x + 1) = 0$ . The value $x = 2$ makes a denominator $0$ , so it isn't a solution. The only solution is $x = -1$ .

The value $x = 2$ satisfies the equation after the fractions are cleared, but it makes $x - 2 = 0$ in the original denominators, so it is extraneous.

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