We offer top techniques on how to solve rational inequalities for high school students. It is an important part of algebra and calculus. Of course, there are a lot of methods to solve these inequalities. Some of them are based on advanced tools in analysis while others are mainly based on the simplification of the expression of the rational equation.

**Basics of Rational Fractions**

**numerator**and $b$ is the

**denominator.**Then fraction is a numerator/denominator. For details on computation for this class of numbers, we refer to the fractions chapter.

**Sign of a fraction**

**Positive fraction:**The fraction $\frac{a}{b}$ is positive if its numerator and denominator have the same sign, i.e. positive together or negative together. This means that $a$ and $b$ are positive, or $a$ and $b$ are negative.

**Negative fraction:**The fraction $\frac{a}{b}$ is negative if the numerator and denominator have opposite signs. This means that $a$ is positive and $b$ is negative or $a$ is negative and $b$ is positive.

**The first-order rational functions**

**Example I:**Solve the rational inequality

**Proof:**As the fraction is positive then we have $x-2\ge 0$ and $x+1>0$ or $x-2\le 0$ and $x+1<0$. This means that $$x\ge 0\quad\text{and}\quad x>-1,$$

**Example II:**Determine the solution of the following rational inequality $$\frac{3x-2}{x+1}\ge 2.$$ This problem can also be reformulated as $$ \frac{3x-2}{x+1}-2\ge 0.$$

**Rational inequalities with quadratic functions**

**Note:**The same technique can also be applied for rational inequalities defined by the polynomial functions with higher degrees.