SOLUTION: Find all values of x such that \frac{x}{x - 5} = \frac{4}{x - 4} + 2x + 3

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Question 1209166: Find all values of x such that
\frac{x}{x - 5} = \frac{4}{x - 4} + 2x + 3
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


x%2F%28x-5%29=4%2F%28x-4%29%2B2x%2B3

The first step in the algebraic process is clear: multiply everything on both sides by the least common denominator, which is %28x-5%29%28x-4%29.

x%28x-4%29=4%28x-5%29%2B%282x%2B3%29%28x-4%29%28x-5%29

Without going any farther with the algebraic solution, we can see this will give us a cubic equation. There is a method for finding the exact solutions of a cubic equation. But that method is so awkward that asking us to use it to find the exact solutions is unreasonable.

So if you want to do that, go ahead. But I would just graph the original two equations using something like the graphing utility on desmos.com to find the approximate solutions. Doing that gives two values of x, which are (approximately)

ANSWERS:
-1.01684
3.69777

Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.

When for cubic equation we found two real roots,
we should find and show the third real root, for completeness.

ANSWERS:
x = -1.01684,
x = 3.69777,
x = 5.31907.

Solved using graphing calculator www.desmos.com/calculator
which is free of charge Internet calculator for common use.