5. Solving Rational Equations
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You can use algebraic tools to solve the equation for x. Could you graph both functions and look for their intersections?
See solution.
To solve it algebraically, one of the first steps we could do is to get rid of the denominators. To do that, we find the least common denominator of the rational expressions and multiply both sides by it. (q(x)s(x))p(x)/q(x) = r(x)/s(x)(q(x)s(x)) ⇓ p(x)s(x) = r(x)q(x) Once we have gotten rid of the denominators, we obtain a polynomial equation which we can solve by factoring. Finally, we must check the solutions into the original equation to be sure we have not gotten extraneous solutions — solutions that do not actually solve the original equation.
To solve the equation graphically, we graph both y=f(x) and y=g(x) on the same coordinate plane and look for the intersections between the graphs.
To solve a rational equation numerically, we must guess what could be the solution and substitute some values of x into both functions. In this case, making a table is an excellent idea to organize the data.
| x | f(x) | g(x) |
|---|---|---|
| a | f(a) | g(a) |
| b | f(b) | g(b) |