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Class 11 Revision Notes Linear Inequalities

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Course

Mathematics – Iii

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MATS University

Academic year: 2023/2024
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Revision Notes

Class - 11 Mathematics

Chapter 6 - Linear Inequalities

● An inequality is a relationship that exists between two values that aren't equal.

● For example, x > 9. Here there is a relation between x & 9.

● Any two algebraic expressions or real numbers related by symbol ‘ < ’, ‘ > ’, ‘  ’ or ‘  ’ form inequality. ● Inequalities can be used to solve problems in science, mathematics, statistics, economics, optimization problems, psychology, and other fields.

Example of inequality in daily life: Rina and Samira have Rs. 5,000 & wants to buy t-shirts and shoes for trekking. The cost price of t-shirt and shoes is Rs. 250 and Rs. 550 respectively. We can write the above statement mathematically using inequalities, as follows;

Let the number of t-shirt they can buy be x & number of shoes be y.

Then, the total amount spent by them is 250x + 550y  5000

Here, the total amount is upto Rs. 5,000. The above given statement consists of two statements as, 250x + 550y  5000 which is an inequality and 250x + 550y  5000 is an equation

Notations: ● The notation a < b means, a is less than b. ● The notation a > b means, a is greater than b. ● The notation a  b means, a is not equal to b. ● The notation a  b means, a is less than or equal to b. ● The notation a  b means, a is greater than or equal to b.

Types of Inequalities: ● Numerical inequalities: Relationship between numbers. For example, 8 < 19 ● Literal or variable inequalities: Relationship between variables or between a variable and number. Example, x < 19 ● Double Inequalities: Relationship from two side. For example, 25 < x < 19 ● Strict inequalities: An inequality that employs symbols < or > The symbols  and  are not used.

For example, y < 4; 1 < 4

● Slack inequalities. An inequality that employs symbols  or . Example, y  7

● Linear inequalities in one variable: A one-variable inequality involving a linear function. Example, y < 4

● Linear inequalities in two variables: An inequality involving a two-variable linear function. Example, 5x+ 7y > 4

● Quadratic inequalities: An inequality which employs a quadratic function. Example, 7x 2  3x  4

Solution for linear inequality in one variable: Solution & Solution Set:

Case 3 : If x is a real number, then solution set is

 
  
3
, .

By representing the case 3 solution on a number line, we get

Question: Solve7x  2  5x  8. Show the graph of the solutions on number line.

Ans: By subtracting 2 from both side, we get 7x  5x  6

By subtracting 5x from both side, we get2x  6

On dividing 2 both side, we get x  3

We can represent this in Number line below.

(Above Number line is drawn by using Paint)

Graphical Solution of Linear Inequalities in 2 variables:

● The Cartesian plane is divided into two equal sections by a line. ● Each component is referred to as a half plane. ● A non-vertical line divides the plane into lower and upper half planes, while a vertical line divides it into left and right half planes. ● In the Cartesian plane, a point will either lie on a line or in one of the half planes. ● The solution zone is the area that contains all of the solutions to an inequality.

(All Graphs are drawn with the help of GeoGebra and Paint) ● To find the half plane represented by an inequality, simply choose any point

 a, b (not online) and see if it meets the inequality.

● If it does, the inequality represents the half plane and shades the region that contains the point; if it does not, the inequality represents the half plane that does not contain the point.

● For convenience, the point  0,0is preferred.

● Example: x + 2y  9

Ans:

Question:

Solve the following system of inequalities graphically5x + 4y  40 ,x  2 and

y  3.

Ans:

Step 1 : Draw lines for5x + 4y  40 ,x  2 and y  3.

Step 2 : For each of these linear inequalities, find the solution zone.

Step 3 : Locate a common area. The solution region is a common region.

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Class 11 Revision Notes Linear Inequalities

Course: Mathematics – Iii

39 Documents
Students shared 39 documents in this course

University: MATS University

Was this document helpful?
Class XI Maths 1
Revision Notes
Class - 11 Mathematics
Chapter 6 - Linear Inequalities
● An inequality is a relationship that exists between two values that aren't equal.
● For example,
x > 9
. Here there is a relation between
x
&
9
.
● Any two algebraic expressions or real numbers related by symbol ‘
<
’, ‘
>
’, ‘
’ or ‘
’ form inequality.
● Inequalities can be used to solve problems in science, mathematics, statistics,
economics, optimization problems, psychology, and other fields.
Example of inequality in daily life:
Rina and Samira have
Rs. 5,000
& wants to buy t-shirts and shoes for trekking.
The cost price of t-shirt and shoes is
Rs. 250
and
Rs. 550
respectively. We can
write the above statement mathematically using inequalities, as follows;
Let the number of t-shirt they can buy be
x
& number of shoes be
y
.
Then, the total amount spent by them is
250x + 550y 5000
Here, the total amount is upto
Rs. 5,000
.
The above given statement consists of two statements as,
250x + 550y 5000
which is an inequality and
250x + 550y 5000
is an equation
Notations:
● The notation
a < b
means,
a
is less than
b
.
● The notation
a > b
means,
a
is greater than
b
.
● The notation
a b
means,
a
is not equal to
b
.
● The notation
a b
means,
a
is less than or equal to
b
.
● The notation
a b
means,
a
is greater than or equal to
b
.

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