Simple Inequalities (Linear and Quadratic)

Overview

Inequalities represent a relationship between quantities, stating that one is greater than, less than, or equal to another.
Simple inequalities are part of the wider spectrum of inequalities and include linear and quadratic inequalities.

A linear inequality looks much like a linear equation, but with an inequality sign instead of an equals sign.
Linear inequalities can be solved in a similar way to linear equations.
Examples of linear inequalities include x + 3 > 7 or 2y - 4 ≤ 8.

Quadratic inequalities are inequalities that involve a quadratic expression, such as x^2 + 3x - 10 > 0.
The graph of a quadratic inequality is a parabola which can be useful in understanding and solving quadratic inequalities.

To solve a simple inequality, the aim is to isolate the variable.
Basic operations such as adding, subtracting, multiplying, and dividing can be performed on both sides of an inequality, much like in an equation.
When multiplying or dividing by a negative number, the direction of the inequality sign must be reversed.
For quadratic inequalities, solutions are often found by first solving the corresponding quadratic equation, and then deciding on the appropriate range for the variable.

Inequalities can be used to represent various real-world situations where there is a range of possible solutions.
For example, inequalities can be used to model situations involving budgets, limits, or any situation where there may be a minimum or maximum limit.

Always remember to flip the inequality sign when multiplying or dividing by negative numbers.
When solving quadratic inequalities, remember that the solution may be a range of values, not a specific value.
Avoid ambiguity by always using clear notation, especially when dealing with compound inequalities.

Try out numerous problems involving linear and quadratic inequalities to hone your skills.
Problems should include both algebraic and word problems to provide a wide range of practice.
Solving inequalities should involve performing operations correctly, using clear notation, and correctly interpreting inequality statements. Be assured that practice in this area will build a solid foundation in understanding and solving inequalities.