We are given the rational equation x over (x minus 4) equals 2. A rational equation has a variable in the denominator. Our strategy is to clear the fraction by multiplying both sides by the denominator, then solve the resulting linear equation. We must also check for extraneous solutions.
To clear the fraction, we multiply both sides of the equation by the denominator (x minus 4). On the left side, the (x minus 4) terms cancel out, leaving just x. On the right side, we have 2 times (x minus 4). This gives us the equation x equals 2 times (x minus 4). Note that we can only do this if x is not equal to 4, because that would make the denominator zero.
Now we solve the linear equation x equals 2 times (x minus 4). First, distribute the 2 on the right side to get x equals 2x minus 8. Next, subtract 2x from both sides to get x minus 2x equals negative 8. This simplifies to negative x equals negative 8. Finally, divide both sides by negative 1 to find x equals 8.
We found x equals 8. Now we must verify this solution by substituting it back into the original equation. We replace x with 8 in x over (x minus 4) equals 2. This gives us 8 over (8 minus 4), which simplifies to 8 over 4, and then to 2. Since 2 equals 2 is a true statement, our solution is correct. Also, x equals 8 does not make the denominator zero, so it's not an extraneous solution.
Let's consider the domain of the original equation. The expression x over (x minus 4) is undefined when the denominator is zero, which happens when x equals 4. This means x cannot be 4. On the graph of y equals x over (x minus 4), there's a vertical asymptote at x equals 4. Our solution x equals 8 is not equal to 4, so it's within the domain. When we substitute x equals 8, we get the point (8, 2) on the graph, which lies on the curve. Therefore, x equals 8 is the complete and valid solution to our rational equation.