Section 7-5
Solving Rational Equations
So far in this chapter we've just dealt with rational expressions - how to
simplify them, multiply them by one another, divide them. Rational equations
are equations to be solved.
CROSS MULTIPLYING
One technique for solving rational equations is cross-multiplication — what
some textbooks call the means/extremes property.
Example:
Solve:
Cross multiplying, we get:
x2 – 2x = 3 x + 6
This quadratic equation can be solved by factoring.
x2 – 5x – 6 = 0
(x + 1)(x – 6) = 0
Remember to check in the original equation for validity of solutions. In this
case, both x = –1 and x = 6 are valid.
LEAST COMMON DENOMINATORS
Another method is to multiply through by a least common denominator.
Example:
Solve:
The least common denominator (LCD) in this case is 16x. So, multiply
both sides of the equation by 16x.
x2 – 6 = 5x
Solve the quadratic equation by factoring.
x2 – 5x – 6 = 0
(x – 6)(x + 1) = 0
x = 6 or x =
–1.
Remember to check back to make sure these solutions are valid – that
is, that they don't result in division by zero when substituted in the original
equation. In this case, both solutions are valid.
