Question

lesson 2.5 review - solving rational equations
lcd = (x + 4)(x - 2)
③ \\(\frac{x}{x - 2} - \frac{2}{x + 4} = \frac{12}{x^2 + 2x - 8}\\)
① factor denominators
② find lcd
③ multiply to create lcd
④ solve using numerator
⑤ check for extraneous solutions!
\\(\frac{x}{(x - 2)} - \frac{2}{(x + 4)} = \frac{12}{(x + 4)(x - 2)}\\)
\\(\frac{x}{(x - 2)} \cdot \frac{(x + 4)}{(x + 4)} - \frac{2}{(x + 4)} \cdot \frac{(x - 2)}{(x - 2)} = \frac{12}{(x + 4)(x - 2)}\\)
18 multiple - choice question
copy that move.
i love this method as she shows because again,
only the numerators will matter once everyone
has the lcd.
that means the next line of work will become
\\(x(x + 4) - 2(x - 2) = 12\\)
\\(x(x + 4) + 2(x - 2) = 12\\)
\\(x(x + 4) - 2(x - 2) = 12(x + 4)(x - 2)\\)
\\(x(x + 4) - 2(x - 2) = 0\\)

Explanation:

Step1: Recall solving rational equations

To solve \(\frac{x}{x - 2}-\frac{2}{x + 4}=\frac{12}{x^{2}+2x - 8}\), first factor denominators. The denominator \(x^{2}+2x - 8\) factors to \((x + 4)(x - 2)\), so \(LCD=(x + 4)(x - 2)\).

Step2: Multiply each term by LCD

Multiply each term by \((x + 4)(x - 2)\):

• For \(\frac{x}{x - 2}\), multiply by \((x + 4)(x - 2)\) gives \(x(x + 4)\) (the \((x - 2)\) cancels).
• For \(-\frac{2}{x + 4}\), multiply by \((x + 4)(x - 2)\) gives \(-2(x - 2)\) (the \((x + 4)\) cancels).
• For \(\frac{12}{(x + 4)(x - 2)}\), multiply by \((x + 4)(x - 2)\) gives \(12\) (the \((x + 4)(x - 2)\) cancels).

So the equation becomes \(x(x + 4)-2(x - 2)=12\).

Answer:

A. \(x(x + 4)-2(x - 2)=12\) (assuming the first option is labeled A, with the equation \(x(x + 4)-2(x - 2)=12\))