SOLUTION: Please help! Solve and check, if no solution is possible, say so. {{{ (5)/(4x)+2=(1)/(x) }}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please help! Solve and check, if no solution is possible, say so. {{{ (5)/(4x)+2=(1)/(x) }}}      Log On


   

Question 622934: Please help! Solve and check, if no solution is possible, say so.
+%285%29%2F%284x%29%2B2=%281%29%2F%28x%29+
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+%285%29%2F%284x%29%2B2=%281%29%2F%28x%29+
There are different ways to solve equations where the variable is in one or more denominators. The way I prefer is this:
  1. Find the lowest common denominator (LCD) of all the denominators on both sides of the equation.
  2. Multiply both sides of the equation by the LCD found in step one. If this is done correctly, all the fractions will disappear.
  3. Use appropriate techniques to solve whatever equation you now have.
  4. Check your solution(s). For each solution, you must make sure that no denominators turn out to be zero. Reject any "solution" that makes a denominator zero.
Let's see how this works on your compound fraction.

1. Find the LCD.
Your denominators are 4x and x. The LCD of these is 4x.

2. Multiply both sides of the equation by the LCD:
%284x%29%28+%285%29%2F%284x%29%2B2%29=%284x%29%28%281%29%2F%28x%29%29+
On the left side we need to use the Distributive Property:
%284x%29%2A%285%2F%284x%29%29%2B%284x%29%2A%282%29=%284x%29%28%281%29%2F%28x%29%29+
All the denominators cancel out:
1%2A5+%2B+4x%2A2+=+4%2A1

3. Solve.
Solving usually starts with simplifying both sides of the equation.
5+%2B+8x+=+4
This is a linear equation which we should know well how to solve:
8x+=+-1
x+=+-1%2F8

4. Check.
Use the original equation to check:
+%285%29%2F%284x%29%2B2=%281%29%2F%28x%29+
Checking x = -1/8:
+%285%29%2F%284%28-1%2F8%29%29%2B2=%281%29%2F%28%28-1%2F8%29%29+
We can already see that neither denominator will turn out to be zero. So we have completed the required check. (You are welcome to finish the check to see if we made a mistake.)

P.S. I suspect that you posted several other similar problems. I hope this helps you figure out how to do them, too.