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A fundamental inequality in geometry stating that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side, such as $a + b > c$ for the sides of a triangle.

An inequality akin to the equation of an ellipse but with an inequality sign. For example, $\frac{x^2}{a^2} + \frac{y^2}{b^2} < 1$ represents the points inside an ellipse.

An inequality involving a quadratic expression, for example, $x^2 - x - 6 > 0$.

A set of two or more linear inequalities in the same variables, such as $\begin{cases} y > 2x + 1 \\ y \leq -x + 3 \end{cases}$.

An inequality containing a ratio of polynomials, such as $\frac{x-1}{x+2} \leq 0$. Critical points from the numerator and denominator are used to find the solution set.

A combination of inequalities that can involve linear, quadratic, rational, or other nonlinear entities, such as $\begin{cases} y > x^2 + 1 \\ x - y < 3 \end{cases}$.

An inequality involving the absolute value, such as $|2x + 1| > 3$. It requires considering both the positive and negative scenarios of the expression inside the absolute value.

An inequality that combines two or more simple inequalities using the words 'and' or 'or', for example, $3 < x + 1 < 5$ means $x$ can take on values between $2$ and $4$.

An inequality involving logarithmic functions, like $\log(x) > 1$.

An inequality related to hyperbolas, similar to a hyperbolic equation, for example, $\frac{x^2}{a^2} - \frac{y^2}{b^2} > 1$, which indicates the region outside the hyperbola.

An inequality involving a linear expression in one variable without exponents, such as $3x - 1 > 2$.

An inequality containing an exponential function, such as $2^x < 8$.

An inequality defined by different expressions depending on the interval the variable lies in, such as $f(x) = \begin{cases} x^2, & x \leq 0 \\ -x, & x > 0 \end{cases}$, then solving for $f(x) < 1$.

An inequality containing a polynomial, such as $x^2 - 4x + 3 < 0$.