Rules, Symbol, Graphical Representation of Linear Inequalities
In algebra, inequalities are mathematical statements describing the relationship between two expressions with an inequality symbol. The terms on both sides of the inequality sign are not the same. It means that the formula on the left must be greater than or less than the formula on the right, or vice versa. If the relationship between two algebraic expressions is defined using inequality symbols, we will discuss literal inequalities. Linear inequalities are the expressions that compare values using inequality symbols such as ‘<’, ‘>’, ‘≤’ or ‘≥’. These values could be numbers or algebra, or a combination of both. For example, 9<11, 21>17 are examples of numerical inequalities, and x>y, y<19-x, x ≥ z > 11 are examples of algebraic inequalities (also known as literal inequalities). In this article, we will also discuss linear inequalities graphical representation.
Symbols that represent Inequalities are:
- > (greater than)
- < ( less than)
- ≥ ( greater than or equal to)
- ≤ ( less than or equal to)
- ≠ ( not equal to)
The symbols ‘ < ‘ and `>` represent strict inequalities, and the` ≤` and `≥` represent loose inequalities. Linear inequalities look like linear equations with the inequality sign replaced by the equal sign.
Rules of Inequalities
Here are some examples of inequalities:
Rule 1: If the inequalities are related, you can skip the intermediate inequalities.
- If, a < b and b < c, then a < c
- If, a > b and b > c, then a > c
Example: If Akarsh is older than Aman and Aman is older than Abhishek, then Akarsh must be older than Abhishek.
Rule 2: Only one of the following can be true: a=b, a > b, a < b
Example: If Aman has more money than Akarsh ( a > b ). So, Aman does not have less money than Akarsh ( not a < b ), and Aman does not have equal money as Akarsh ( not a = b )
Rule 3: Adding the number p to both sides of an inequality. If a < b, then a p < b p
Example: Aman has less money than Akarsh. If both Aman and Akarsh Rs.5 more, the Aman will still have less money than Akarsh.
Just like that:
- If a < b, then a − p < b − p
- If a > b, then a p> b p, and
- If a > b, then a − p> b − p
Rule 4: Multiplying the numbers a and b by a positive number does not change the inequality. Multiplying both a and b by a negative number reverses the inequality. Multiplying by (-2), a < b becomes b < a.
- If a< b, and p is positive, then ap< bp
- If a< b, and p is negative, then ap> bp(inequality swaps)
Rule 5: Preceding p and q with a minus sign changes the direction of the inequality.
- If a< b, then −a > −b
- If a > b, then −a < −b
- It is the same as multiplying by (-1) and changes direction.
Rule 6: Taking the reciprocal 1 / value of p and q changes the direction of the inequality. If both a and b are positive or both are negative:
- If, a < b, then 1/a > 1/b
- If a > b, then 1/a < 1/b
Rule 7: The square of the number is always greater than or equal to zero a2 ≥ 0
Example: (3)2= 9, (−3)2 = 9, (0)2 = 0
Rule 8: Taking a square root will not change the inequality.
Example:
a=2, b=7
2≤7,√2≤√7
Rules for solving Inequalities
The rules for solving inequalities are similar to the rules for solving linear equations. However, there is one exception when multiplying or dividing by a negative number. To resolve the inequality, you can:
- On both sides, add the same number.
- From both, the sides subtract the same number.
- Multiply both sides by the same positive number.
- Divide both sides by the same positive number.
- Multiply the same negative number on both sides and reverse the sign.
- Divide the same negative number into both sides and reverse the sign.
How to Graph Linear Inequalities?
Linear inequality graphs typically use boundaries to divide the coordinate plane into two regions. Part of the region consists of all the solutions to inequality. The border is drawn with a dashed line representing `>` and ‘ < ‘ and a solid line representing `≥` and `≤`.
To graphically represent an inequality:
- If inequality exists, target y for the expression. Example: y> x 2
- Replace the inequality sign with an equal sign and select either the y or x value.
- Plots and line graphs of these arbitrary values for x and y.
- Remember to draw a solid line if the inequality symbol is ≤ or ≥ and a dashed line if it is.
- If the inequality > or ≥ and < or ≤ shade the top and bottom of the line.
Important Result
- If a, b ∈ R, and b ≠ 0, then
(i) ab > 0 or a b > 0 ⇒ a and b are of the same sign.
(ii) ab < 0 or a b < 0 ⇒ a and b are of opposite sign.
- If a is any positive real number, i.e., a > 0, then
(i) | x | < a ⇔ – a < x < a
| x | ≤ a ⇔ – a ≤ x ≤ a
(ii) | x | > a ⇔ x < – a or x > a
| x | ≥ a ⇔ x ≤ – a or x ≥ a
Conclusion: Hope you have understood some basic information regarding Linear Inequalities and linear inequalities graphical representation.
Frequently Asked Questions
1.What are the various symbols of inequality?
The various inequalities in mathematics are:
- Inequalities (≠)
- Strict inequalities (>, < )
- Slack inequalities (≥, ≤)
2.Mention the nature of inequality.
The properties of the inequality are: Addition and subtraction properties, Multiplication and division properties, Transition properties, Converse properties.
3.Does exchange the left, and right values change the direction of the inequality?
Yes, swapping the left and right values will change the direction of the inequality.
4.Name the rule for solving the inequality.
The following rule does not affect the direction of the inequality:
- Add or subtract the same number on both sides of the inequality
- Multiply or divide the same positive number on both sides of the inequality
- Simplify one side of the equation if possible
5.What is inequality?
When two real numbers or algebraic expressions are related to the symbols>, <, ≥, ≤, the relationship is inequality.
