Solution of Linear and Quadratic Inequalities

Introduction to Linear and Quadratic Inequalities

Inequalities extend the concept of equations by introducing ‘greater than’ (>) and ‘less than’ (<) conditions.
These inequalities can exist between two algebraic expressions.
Solutions to inequalities are ranges of values that satisfy the given conditions.

Linear inequalities are solved in a manner very similar to linear equations.
The aim is to isolate the variable on one side of the inequality.
Add, subtract, multiply, or divide both sides just as in equations, but be aware that multiplying or dividing by negative numbers reverses the inequality direction.
For example, to solve the inequality 2x + 3 < 7, subtract 3 from both sides to get 2x < 4, then divide by 2 to get x < 2.

When dealing with quadratic inequalities, start by rearranging the inequality to one side i.e., in the form of ax^2 + bx + c < 0 or ax^2 + bx + c > 0.
Factorise or complete the square to find ‘critical values’ where the expression equals zero.
Take ranges between critical values and test a value in each range by substituting it into the factorised inequality.
For example, to solve (x - 1)(x - 3) > 0, the critical values are x = 1 and x = 3. Testing gives the solutions x < 1 or x > 3.

Solutions to the inequalities can be represented on a number line.
The solutions to quadratic inequalities are typically ranges of values and can be represented on a number line with hollow or filled circles indicating whether the endpoints are included in the solution.

Remembering these key formulas and rules will enable you to tackle linear and quadratic inequalities comfortably and effectively.