How many solutions does the equation xx - 3 + 2 = 3x - 3 have?

SAT Nonlinear equations in one variable and systems of equations in two variables Practice Question 18 (Hard)

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Question bank / Nonlinear equations in one variable and systems of equations in two variables

Math · Advanced MathHard

How many solutions does the equation $\frac{x}{x - 3} + 2 = \frac{3}{x - 3}$ have?

Zero

Exactly one

Exactly two

Infinitely many

Show answer and explanation

Answer: A

Multiply both sides by $x - 3$ : $x + 2(x - 3) = 3$ , so $3x - 6 = 3$ and $x = 3$ . But $x = 3$ makes the denominators zero, so it is not allowed. The equation has no solutions.

Choice B accepts $x = 3$ without checking it against the original equation, where it makes the expressions undefined.

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