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Karnataka Class 11 Commerce Maths Linear Inequalities

Karnataka Class 11 Commerce Maths Linear Inequalities

Karnataka Class 11 Commerce Maths Linear Inequalities

Karnataka Class 11 Commerce Maths Linear Inequalities : “Larl Friedrich Gauh” a French Mathematician developed the theory of
numbers. In this chapter we consider the expression, which involve signs such as ‘<’ (less than), ‘>’ (greater than) ‘≤’ (less than or equal) and ≥ (greater than or equal) called Inequalities and such expressions is generally called “Inequations’.

In this chapter you will study linear inequalities in one and two variable. The study of inequalities is very useful in solving problems in the field of science, mathematics, statistics, optimization problems, economics etc., It plays a very important role in Linear programming problems.

Karnataka Class 11 Commerce Maths Linear Inequalities

Definition of Inequality:

If two real numbers or two algebraic expression related by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form an Inequality

Examples:

3< 5, 10>8, 5x<6, 4x – 3 < 8, 3x – 7>0, 4x ≤ 7, 7x – 5 ≥ 0, y ≤ 4, 2x + 4y ≤ 7, x – 4y ≥ 10 etc.,

(1). The Inequality in the form 3< 5, 7> 5, are the example of numerical inequalities.

(2). The Inequality in the form x < 5, y > x > 3, y < 7 etc. are called literal inequalities.

(3). The inequality in the form 3 < 5 < 7 (read as 5 is greater than 3 and less than 7)

3 < x < 5 (read x is greater than or equal to 3 and less than 5)

2 < y < 4 (y is greater than 2 and less than or equal to 4) are the example of double inequalities.

(4). Inequalities in the form: ax + b < 0

ax + b > 0
ax + by < c
ax – by > c are strict inequalities.

(5). Inequalities in the form: ax + b < 0

ax – b < 0
ax + by > c
ax – by < c, are slack inequalities.

(6). inequalities in the form: ax + b < 0
ax – b < 0
ax + b > 0
ax + b > 0 (a ≠ 0) are, inequalities in one variable ‘x’

(7). Inequalities in the form: ax + by < c
ax + by > c
ax + by < c
ax – by > c , (a ≠ 0, b ≠ 0) are inequalities in two variable x & y

(8). Inequalities in the form: ax2 + bx + c < 0 (a ≠ 0, b ≠ 0) and ax2 + bx + c > 0 are quadratic inequalities in one variable x

Karnataka Class 11 Commerce Maths Linear Inequalities

SOME IMPORTANT SETS:

N = Set of Natural numbers = {1, 2, 3, ……..}
W = Set of whole numbers = {0, 1, 2, 3 ……..}
I or Z = Set of Integers = {0, 1, +2, + 3 ……..}
I+ = Set of +ve Integers = {1, 2, 3 ……..}
I- = Set of –ve integers = {-1, -2, -3, ……..}
Q = set of rational numbers = {p/q, p,q∈ I, q ≠ 0 }
R = Set of real numbers = {all Natural, whole, Integer, rational Irrational numbers}
R+ = Set of +ve Real Numbers
R- = Set of –ve Real numbers

Karnataka Class 11 Commerce Maths Linear Inequalities

Linear Inequalities in one variable:

A linear inequality in one variable is an expression of the form ax + b < c, ax+b>c, ax + b≤ c or ax + b ≥ c where a, b, c ∈ R (set of Real numbers and a ≠ 0).
Example 5x + 3 < 4, 4x – 5 ≥ 7, 8x – 1 ≤ 3, 3x + 1 < 7, are linear inequations.

In the above examples we can find the values of ‘x’ which makes the above in equality a statement. True values of x are called
solutions of inequality.

Solution set : A solution set of inequality is the set of all real numbers (R) that satisfies the inequality the method of finding the solution set of the inequality is known as solving the inequation

Karnataka Class 11 Commerce Maths Linear Inequalities

Rules for solving the linear inequation:

Rule 1 : The inequality does not change if the same number is added on both the side of the inequality.

Rule 2: (Multiplication or Division Rule) The inequality does not change if we multiply or divide both the side of inequality by the same +ve real number.

Rule 3: The inequality reverses its direction if we multiply or divide both the side of the inequality by the same –ve real number.
i.e. If a < b ⇒ a + K < b + K (K ⇒ R)
a > b ⇒ a + K > b + K (K ⇒ R)
a > b ⇒ aK < bK (K < 0 i.e. K is –ve)

NOTE:

‘O’ Represent Solution Exclude
‘•’ Represent include
∞ Infinity
( ) both side excludes
( ] left excludes and right include
[ ) Left Include and Right exclude
[ ] both side include.

Karnataka Class 11 Commerce Maths Linear Inequalities

System of Linear Inequalities in one variable

Two or more linear inequalities in one variable together form a system of linear inequalities in one variable. The solution set of the system is the set of all values which satisfy all the inequation involved in the system. i.e., the Intersection of the solution set of each is the solution of the system of linear inequation.

Karnataka Class 11 Commerce Maths Linear Inequalities

Linear Inequation in two variable and their graphical representation:

A linear inequality in two variable is an expression in the form ax + by < c, ax + by > c, ax + by ≤ c or ax + by ≥ c (a, b, c, ∈ R and a ≠ b ≠ 0)

Example: 4x – y < 3, 2x + y > 5, x + 3y ≤ 3, 2x – y ≥ 1

Solution Set: The solution set for a linear inequality in two variable is the set of all values of (x, y) which satisfy the given linear inequality.

Geometrically this represents a section of co-ordinate plane described by a set of co-ordinate system with x and y axes

Working rule to finding the solution set of linear inequalities in two variables

Step 1: Replace the inequality sign involving in the given statement by Equality sign. The resulting equation represents a straight line, which acts as the boundary of the solution.

Step 2: Draw the line graph of the equation from step 1 using convenient point on the line and join them. If the inequality in the given equation is

(i) < or > then the boundary line is not included in solution set and represent the boundary by broken line.

(ii) ‘≤’ or ‘≥’ then the boundary line is included in solution set and represent the boundary by solid line.

Step 3: The line in step 2 divides the co-ordinate place into two region one above the boundary and the other below. (or is the boundary line is vertical then one region is to left and the other is right). To find the required region choose any convenient point which is not on the boundary and verify whether the coordinate of this point satisfy the given inequality or not. If it satisfy the inequality then the solution set is the set of all point which lie on the same side of the boundary as the chosen point. If the inequality does not satisfy the co-ordinates of the chosen point then the solution set is the set of all points on the opposite side of the boundary.

Karnataka Class 11 Commerce Maths Linear Inequalities

System of Linear Inequation in two variable and their graphical solution.

Two or more linear equalities in two variable together forms a system of linear inequalities in two variable. The solution set is the set which satisfies both the linear inequation.

Example:

i) x ≥ 3, y ≥ 2
ii) 2x + y ≥ 6, 3x + 4y < 12
iii) 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0
iv) x + 2y ≤ 10, x + y ≤ 1, x – y ≤ 0, x ≥ 0, y ≥ 0

here x ≥ 0, y ≥ 0 are condition to the system of inequalities called non negative condition. The solution set of the about system of linear inequalities in two variable is the set of all ordered pain (x, y) which satisfies each inequality of the system simultaneously.

Karnataka Class 11 Commerce Maths Linear Inequalities

Feasible Solution:

A point (x, y) on the co-ordinate plane is called Feasible solution of the system of inequalities in two variable is it satisfy all the in equation of the system.

The set of all feasible solution form a feasible region.

Karnataka Class 11 Commerce Maths Linear Inequalities

Linear Inequalities

Linear inequalities, Algebraic solutions of linear inequalities in one variable and their representation on the number line and examples.

Linear inequality includes a linear function and an inequality. An inequality is obtained when two expressions are connected by the sign of ‘greater than’ or ‘less than’. It is actually a system with just two or more inequalities together. Linear inequality is similar to the linear function if the sign of linearity replaces the symbol of inequality.

Consider a function f (y), then relation of real numbers ’n’ with function can be represented by inequality as shown below: f (y) < n or f (y) ≤ n. If function f (y) is considered as x 0 + a 1x 1 + a 2x 2 +…+ a nx n < 0 or x 0 + a 1x 1 + a 2x 2 +…+ a nx n ≤ 0 will show linear inequality.

Karnataka Class 11 Commerce Maths Linear Inequalities

Let’s discuss Linear Inequalities Applications:

Collection of solutions of a linear inequality creates a half set of n-dimensional space.

One of the inequalities will respond to linear equation and this set or collection also responds to intersection of half – planes which are explained by use of individual inequality. Half – sets are convex sets. Intersection of a collection of convex sets (or half sets) is also convex. Let’s consider a system of ‘n’ equality constraints in ‘m’ positive variables.

Here equality constraints are linearly independent. Y = c, Y ≧0. Suppose we are given a set of ‘n’ column vectors of ‘B’ which are linearly independent. These column vectors are known as special column vectors.

To develop an algorithm to determine whether there exists a feasible basis which contains all special column vectors as basic column vectors and to find such a basis if one exists, we use the concept of linear inequality. Such algorithms are used in several applications. This is all about Applications of Linear Inequalities.

Linear equation is an algebraic expression having both constants and variables. Linear expression is an expression which contains (=) sign and highest degree is one, like x +2, x + 5y etc. Number of variables in an equation may be one or more than one. It is possible to convert a non – linear equation into linear equation. Here we will discuss linear equation application.

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