Question

lesson 2.5 review - solving rational equations
③ \\(\frac{x}{x - 2} - \frac{2}{x + 4} = \frac{12}{x^2 + 2x - 8}\\)
\\(\frac{x}{(x - 2)} - \frac{2}{(x + 4)} = \frac{12}{(x + 4)(x - 2)}
\frac{x}{(x - 2)(x + 4)} - \frac{2}{(x + 4)(x - 2)} = \frac{12}{(x + 4)(x - 2)}
① factor denominators
② find lcd
③ multiply to create lcd
④ solve using numerator
⑤ check for extraneous solutions!
multiple-choice question
look at the third fraction (the one whose numerator is 12).
why was it not necessary to multiply it top and bottom by another factor?
□ because the third fraction only has a number in it’s numerator.
□ because the third fraction already had the lcd.
□ because the third fraction never needs to be multiplied top and bottom.

Explanation:

To solve rational equations, we find the least common denominator (LCD) of the fractions. The third fraction's denominator is factored as \((x + 4)(x - 2)\), which is the LCD of the other two fractions (\(\frac{x}{x - 2}\) has denominator \(x - 2\) and \(\frac{2}{x + 4}\) has denominator \(x + 4\), so LCD is \((x - 2)(x + 4)\)). Since the third fraction already has the LCD as its denominator, we don't need to multiply it by another factor. The first option is incorrect (numerator being a number isn't the reason), and the third option is incorrect (we do multiply other fractions to get LCD, but this one already has it).

Answer:

B. Because the third fraction already had the LCD.