# Rational Inequalities

Rational inequations are solved in a similar way to quadratic inequalities, but keep in mind that the denominator cannot be zero.

1. Calculate the roots of the numerator and denominator.

x − 2 = 0 x = 2

x − 4 = 0 x = 4

2. Represent these values in the real line, bearing in mind that the roots of the denominator have to be open, that is to say, they cannot be equal to zero.

3.Take one point from each interval and evaluate the sign in each:

4. The solution is composed of the intervals (or the interval) that have the same sign as the polynomial fraction.

S = (-∞, 2] (4, ∞)

#### Examples

1

Subtract 2 in the two members and reduce to a common denominator.

Calculate the roots of the numerator and denominator.

−x + 7 = 0 x = 7

x − 2 = 0 x = 2

Evaluate the sign:

S = (-∞, 2) (7, ∞)

2

The binomial squared is always positive, but keep the minus sign before it and the demnominador result will always be negative.

Multiply by −1:

(−-∞ , −1] (1, +∞)

3

(−2 , −1] [1, 2)