Solve for all values of x. (x+3)/(x+9)=(6)/(x) Answer: x=

Cross-Multiply: Cross-multiply to eliminate the fractions. (x + 3)/(x + 9) = 6/x Cross-multiplying gives us: (x + 3) * x = 6 * (x + 9)Distribute Simplify: Distribute to simplify the equation. (x + 3) * x = 6 * (x + 9) x^2 + 3x = 6x + 54Move Terms Set Zero: Move all terms to one side to set the equation to zero. x^2 + 3x - 6x - 54 = 0 x^2 - 3x - 54 = 0Factor Quadratic Equation: Factor the quadratic equation. x^2 - 3x - 54 = 0 (x - 9)(x + 6) = 0Solve for x: Solve for x by setting each factor equal to zero. x - 9 = 0 or x + 6 = 0 x = 9 or x = -6Check Validity: Check for extraneous solutions by plugging the values back into the original equation. For x = 9: (9 + 3)/(9 + 9) = 6/9 12/18 = 6/9 2/3 = 2/3 (Valid solution) For x = -6: (-6 + 3)/(-6 + 9) = 6/(-6) -3/3 = -1 (Invalid solution, as it would result in division by zero in the original equation)

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Algebra 2

Simplify variable expressions using properties

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Q. Solve for all values of

x

\newline

\frac{x+3}{x+9}=\frac{6}{x}

\newline

Answer:

x=

Cross-Multiply: Cross-multiply to eliminate the fractions. $\newline$ $(x + 3)/(x + 9) = 6/x$ $\newline$ Cross-multiplying gives us: $(x + 3) \cdot x = 6 \cdot (x + 9)$
Distribute Simplify: Distribute to simplify the equation. $\newline$ $(x + 3) \times x = 6 \times (x + 9)$ $\newline$ $x^2 + 3x = 6x + 54$
Move Terms Set Zero: Move all terms to one side to set the equation to zero. $\newline$ $x^2 + 3x - 6x - 54 = 0$ $\newline$ $x^2 - 3x - 54 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $x^2 - 3x - 54 = 0$ $\newline$ $(x - 9)(x + 6) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $x - 9 = 0$ or $x + 6 = 0$ $\newline$ $x = 9$ or $x = -6$
Check Validity: Check for extraneous solutions by plugging the values back into the original equation. $\newline$ For $x = 9$ : $\newline$ $(9 + 3)/(9 + 9) = 6/9$ $\newline$ $12/18 = 6/9$ $\newline$ $2/3 = 2/3$ (Valid solution) $\newline$ For $x = -6$ : $\newline$ $(-6 + 3)/(-6 + 9) = 6/(-6)$ $\newline$ $-3/3 = -1$ (Invalid solution, as it would result in division by zero in the original equation)