Rational Inequalities

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

Rational Inequality

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:

Interval

Rational Inequality

Rational Inequality Operations

Rational Inequality Operations

Rational Inequality Operations

Intervals

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

S = (-∞, 2] Unión (4, ∞)


Examples

1Rational Inequality

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

Rational Inequality Operations


Calculate the roots of the numerator and denominator.

−x + 7 = 0      x = 7

x − 2 = 0     x = 2

Evaluate the sign:

Rational Inequality Operations

Rational Inequality Operations

Rational Inequality Operations

Intervals

S = (-∞, 2) Unión (7, ∞)


2Rational Inequality Exercise

Rational Inequality Operations

Rational Inequality Operations

Rational Inequality Operations

Rational Inequality Operations

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

Rational Inequality Operations

Multiply by −1:

Rational Inequality Operations

Intervals

(−-∞ , −1] Unión (1, +∞)

3Rational Inequality

Rational Inequality Operations

Rational Inequality Operations

Rational Inequality Operations

Intervals

[−2 , −1] Unión (1, 2)