In this article we are going to learn the concept of Linear inequalities and Quadratic Inequalities. As the questions on inequalities appear frequently in exams, it will be advisable to understand and cover the topic in a thorough manner. Here we will be discussing linear and quadratic inequalities.
What is an inequality?
In algebra, an inequality has algebraic expressions involving variables and constant terms such that one expression is either less than or greater than the other expression.
Basic rules of operations on inequality:
The above rules are the basic rules which will be used in the solutions of the inequalities.
Solving Linear Inequalities
Inequalities can be of different types like linear inequalities, quadratic inequalities. Let us discuss the methods to solve these inequalities one by one.
Solving Quadratic Inequalities
Quadratic Inequalities:Quadratic inequalities are those inequalities in which the highest power of the variable is 2. Now to solve such inequalities use the following steps:
Example 2: Solve the inequality x2 + x - 28 < 2.
Solution : We have x2 + x - 28 < 2Step I: Make the right hand side equal to zero, we get x2 + x - 30 < 0 Step II: Factorize the equation, we get x2 +x - 30 < 0 => x2 + 6x - 5x - 30 < 0 => x(x + 6) - 5(x + 6) < 0 => (x - 5) (x + 6) < 0 Step III: Equate to zero both the factors to get the critical points, we get x = 5, - 6 Plot these points on the number line, we get Now as our inequality is a negative inequality, so the middle part is our solution. Hence the solution is (- 6, 5).
Note: If the original inequality is x2 +x - 30 > 0, then the solution will follow the same above steps and the solution of the inequality will be (- ∞, - 6) U (5,∞) i.e. all the values for which the graph is showing the positive values.
Master all the concepts of Linear and Quadratic Inequalities by watching this Achievers video on Linear and Quadratic Inequalities
Linear & Quadratic Inequalities: Key Learning
If you still have any doubt regarding any rule or method explained in the article, feel free to post it as comment in the section below.