When solving a rational equation we multiply each side of the equation by the least common multiple of the denominators. Use the factors of the given expression to write the denominators of your equation.
Example Solution: 1/x+3 + 1/x-4 = 1/4(x-4)
Practice makes perfect
Let's begin by considering the following rational equation.
P_1(x)/Q_1(x) + P_2(x)/Q_2(x) = P_3(x)/Q_3(x)
Notice that this equation is just an example. We could have more terms on each side. When solving a rational equation, we multiply each side of the equation by the least common multiple of the denominators.
LCM(Q_1(x),Q_2(x),Q_3(x))In our case, we need to multiply by 4(x+3)(x-4).
LCM(Q_1(x),Q_2(x),Q_3(x)) = 4(x+3)(x-4)
The candidates to be denominators are combinations of the factors involved in the least common multiple. Some possible denominators are shown below.
Q_1(x)
Q_2(x)
Q_3(x)
Combination 1
x+3
x-4
4(x-4)
Combination 2
4(x+3)
x-4
x+3
Combination 3
4
x+3
x-4
The above combinations are just a few. We can make many different combinations. Let's use the first one, and for simplicity consider all the numerators to be 1.
1/x+3 + 1/x-4 = 1/4(x-4)
That way we have written a rational equation that can be solved by multiplying each side by 4(x+3)(x-4).
Solving the Equation
We already know the least common multiple of the denominators. We can multiply each side by it and then solve the resulting equation for x.
