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)}\\)
1 factor denominators
2 find lcd
3 multiply to create lcd
4 solve using numerator
5 check for extraneous solutions
multiple - choice question
what will be the lcd we create for this equation?
the lcd has to be (x + 4)(x + 2)
the lcd has to be (x + 4)(x - 2)
the lcd has to be (x - 2), only.
rewatch

Explanation:

To find the least common denominator (LCD) of rational expressions, we first factor all denominators. The denominators here are \(x - 2\), \(x + 4\), and \(x^2 + 2x - 8\). Factoring \(x^2 + 2x - 8\) gives \((x + 4)(x - 2)\) (since \(x^2 + 2x - 8=(x + 4)(x - 2)\) by factoring: looking for two numbers that multiply to -8 and add to 2, which are 4 and -2). The denominators are \(x - 2\), \(x + 4\), and \((x + 4)(x - 2)\). The LCD is the product of the highest power of each distinct factor. The distinct factors are \(x - 2\) and \(x + 4\), so the LCD is \((x + 4)(x - 2)\).

Answer:

The LCD has to be \((x + 4)(x - 2)\)