Solution of Linear Inequality

 Linear inequality definition consists of those expressions one which both sides are not equal. Here, the equal sign is replaced by a greater than symbol, less than symbol, greater than or equal to symbol and less than or equal to symbol. There are different types of inequality as well. They are polynomial inequality, absolute value inequality, rational inequality.

Types of inequalities

Polynomial inequality: This inequality consists of polynomials on both sides of the inequality.

Absolute value inequality: If the inequality consists of absolute functions as well as polynomials, then it is known as absolute value polynomials.

Rational inequality: Inequality consists of a rational term in at least one of the sides is known as rational inequality.

How to solve Linear Inequalities :

Step 1: The inequality must be written in the form of an equation. This is important to simplify the equation.

Step 2: The formed equation is solved for one or more values. Different methods are taken into consideration.

Step 3: All the values are then expressed in the number line.

Step 4: Generally, open circles are used in the number lines to show excluded values of the equation.

Step 5: The interval is found.

Step 6: Now, any random value from the range is taken and placed in the inequation to exhibit whether the value satisfies the inequation.

Step 7: Therefore, the interval will satisfy the inequality and that is the answer.

Graphing of Linear inequality

The plotting graph is quite equal to that of equations. The only difference that exists is that in the case of linear equations the answer can be represented as points but in the case of linear inequality, the answer represents as area. The line is a part of the function and depends on the inequality sign. Steps of plotting the graph of a linear inequality is-

Step 1: The inequation should be arranged in a manner that all they-terms are on or side and x- terms on the other.

For example, y>x-3

Step 2: The graph is drawn based on the equation form of the inequation, i.e. the graph of y=x-3 is plotted on the graph.

Step 3: A solid line should be drawn for greater than or equal to and less than or equal to sign whereas horizontal lines are drawn for greater than and less than sign.

Sign 4: Now, to represent the area of the solution, the part is shaded according to the necessity i.e. if the sign is greater than or greater than- equal to the upper part of the line is shaded. If the sign is less than or less than equal then the lower part of the line is shaded.

Example

• Calculate the range of values of y, which satisfies the inequality: y − 4 < 2y + 5.

Solution:  Add both sides of the inequality by 4.

y – 4 + 4 < 2y + 5 + 4

or, y < 2y + 9

Subtract both sides by 2y.

y – 2y < 2y – 2y + 9

Y < 9 Multiply both sides of the inequality by −1 and change the inequality symbol’s direction. y > − 9

• Solve the inequality 4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )

Solution: Given,

4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )

Or, 4 x + 8 − 1 > 5 − 28 + 7 x

or, 4 x + 7 > − 23 + 7 x

4x + 7 – 7 > -23 + 7x – 7( Performing subtraction on both sides)

or, 4x > -30 + 7x

or, 4x – 7x > -30 + 7x – 7x (Performing subtraction of 7x from both the sides)

or, − 3 x > − 30

Or, -3x (-1) < -30 x (-1) (Multiplying both the sides by -1

Or, 3x < 30

Dividing both the sides by 3, we get;

3x/3 < 30/3

Or, x < 10

Therefore it can be said that x lies between -∞ and 10.

Or, x€ (-∞, 10)

• Solve: 5x – 3 < 3x + 1, where x ∈ R

Solution: Provided, 5x – 3 < 3x + 1

By subtracting 3x from both the sides of the given inequality, 

⇒ (5x – 3) – 3x < (3x + 1) – 3x

⇒ 2x – 3 < 1

By adding 3 on both sides,

⇒ 2x < 4

By dividing both the sides, we get

x < 2

Therefore, x lies between – ∞ and 2 in the number line.

∵ x ∈ R ⇒ x = (- ∞, 2)

Conclusion:

Linear inequalities are those linear functions whose both sides are not equal to each other. Though the inequalities are different from equations, the calculation of inequalities is quite simple and is similar to that of equations. A system of linear inequalities consists of at least two linear inequalities with identical variables. The inequalities can also be solved by plotting graphs.