Linear and Quadratic Inequalities

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.
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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:
  • You can add or subtract same number from both sides of the inequality without changing the truth of the inequality. If a > b, then a+k > b+k e.g. If 7 > 5 then 7 + 3 > 5 + 3 and if 10 < 5 then 10 - 3 < 5 - 3
  • You can multiply or divide both sides of the inequality with the same positive number. It will not change the sign of the inequality. If a>b, then ak > bk ; k > 0 e.g. 8 > 6 => 8 × 4 > 6 × 4
  • If you multiply or divide both sides of the inequality with the same negative number, then the sign of the inequality will change. If a > b, then ak < bk; k< 0 e.g. 9 > 5 => 9 ×( -3) < 5 × (-3)
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.
  • Linear Inequalities Linear inequalities are those inequalities in which the highest power of the variable is 1. These are the simple inequalities to solve. It requires the knowledge of some basic algebraic rules to solve them. Let us take an example.
Example 1: Solve the inequality; (3x+2/-3)< Solution : We have (3x+2/-3)< => 3x + 2 > - 15 (The sign of the inequality has been changed as it is multiplied by - 3) 3x > - 17 => x >(-17/3) is the solution of the given inequality.
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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:
  1. Make right hand side of the inequality equal to zero by transposing the terms (if any) from left hand side.
  2. Factorize the expression on the left hand side and make sure that the coefficient of 'x' is positive in each factor.
  3. Equate to zero both the factors to get the critical points. Plot these points on the number line. There will be three regions on the number line and the rightmost region will give you positive inequalities, middle one will give you negative inequalities and the leftmost part will again give you positive inequalities. You will understand this from the following example.
Example 2: Solve the inequality x2 + x - 28 < 2.
Solution : We have x2 + x - 28 < 2
Step 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.
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Linear & Quadratic Inequalities: Key Learning
  • So here we have discussed how to solve the inequalities. Normally this topic is considered as a difficult one but if you have your basics clear then it is very simple. In case of quadratic inequalities just follow the above given steps and the solution will follow.
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.
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