Solve for all values of x. (x-1)/(x-2)=(6)/(x) Answer: x=

Cross-Multiply: Cross-multiply to eliminate the fractions. (x - 1) * x = 6 * (x - 2)Distribute Simplify: Distribute to simplify both sides of the equation. x^2 - x = 6x - 12Bring Terms Solve: Bring all terms to one side to set the equation to zero and solve for x. x^2 - x - 6x + 12 = 0 x^2 - 7x + 12 = 0Factor Quadratic: Factor the quadratic equation. (x - 3)(x - 4) = 0Set Solve x: Set each factor equal to zero and solve for x. x - 3 = 0 or x - 4 = 0 x = 3 or x = 4Check Valid Solutions: Check for extraneous solutions by plugging the values of x back into the original equation. For x = 3: (3 - 1)/(3 - 2) = 6/3 2/1 = 2 2 = 2 (Valid solution) For x = 4: (4 - 1)/(4 - 2) = 6/4 3/2 = 1.5 1.5 ≠ 6/4 (Invalid solution, as 4 - 2 = 2, and we cannot divide by zero in the original equation)

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Algebra 2

Simplify variable expressions using properties

Full solution

Q. Solve for all values of

x

\newline

\frac{x-1}{x-2}=\frac{6}{x}

\newline

Answer:

x=

Cross-Multiply: Cross-multiply to eliminate the fractions. $\newline$ $(x - 1) \times x = 6 \times (x - 2)$
Distribute Simplify: Distribute to simplify both sides of the equation. $x^2 - x = 6x - 12$
Bring Terms Solve: Bring all terms to one side to set the equation to zero and solve for $x$ . $x^2 - x - 6x + 12 = 0$ $x^2 - 7x + 12 = 0$
Factor Quadratic: Factor the quadratic equation. $\newline$ $(x - 3)(x - 4) = 0$
Set Solve $x$ : Set each factor equal to zero and solve for $x$ . $\newline$ $x - 3 = 0$ or $x - 4 = 0$ $\newline$ $x = 3$ or $x = 4$
Check Valid Solutions: Check for extraneous solutions by plugging the values of $x$ back into the original equation. $\newline$ For $x = 3$ : $\newline$ $\frac{3 - 1}{3 - 2} = \frac{6}{3}$ $\newline$ $\frac{2}{1} = 2$ $\newline$ $2 = 2$ (Valid solution) $\newline$ For $x = 4$ : $\newline$ $\frac{4 - 1}{4 - 2} = \frac{6}{4}$ $\newline$ $\frac{3}{2} = 1.5$ $\newline$ $1.5 \neq \frac{6}{4}$ (Invalid solution, as $4 - 2 = 2$ , and we cannot divide by zero in the original equation)