Question

follow the step-by-step process to solve the rational equation.
a) list all restricted values.
b) determine the lcd of all denominators in the given rational equation.
c) write the new linear equation obtained after multiplying both sides of the rational equation by the lcd
d) solve the equation or state that there is no solution.
\\(\frac{3x}{x^2 - 6x - 7} - \frac{5}{x - 7} = \frac{1}{x + 1}\\)
a) the restricted value(s) is/are \\(x = \square\\).
(use a comma to separate answers as needed )

Explanation:

Step1: Factor the denominator \(x^2 - 6x - 7\)

We factor \(x^2 - 6x - 7\) as \((x - 7)(x + 1)\) because \( -7\times1=-7\) and \( -7 + 1=-6\). So the rational equation becomes \(\frac{3x}{(x - 7)(x + 1)}-\frac{5}{x - 7}=\frac{1}{x + 1}\).

Step2: Find values that make denominators zero

For the denominator \((x - 7)(x + 1)\), setting \((x - 7)(x + 1)=0\) gives \(x - 7 = 0\) or \(x + 1 = 0\), so \(x = 7\) or \(x=-1\).
For the denominator \(x - 7\), setting \(x - 7 = 0\) gives \(x = 7\).
For the denominator \(x + 1\), setting \(x + 1 = 0\) gives \(x=-1\).

Answer:

The restricted values are \(x = -1, 7\)