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Linear Inequalities represent the situation where some constraints are applied. For example, if we say the sum of x and y cannot be greater than 11, then it is written as
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NCERT notes Class 11 Maths Chapter 6 offer well-structured study material to prepare for linear inequalities. It will help the students with quick revision. These notes of Class 11 Maths Chapter 6 are prepared by the Subject Matter Experts according to the latest syllabus provided by CBSE, ensuring that students do not miss out on any key concept. NCERT solutions for class 11 maths and NCERT solutions for other subjects and classes can be downloaded from NCERT Solutions.
A linear inequality is defined as an expression where any two values are compared by the inequality symbol, such as < > ≤ ≥
Let us take an example.
Ram has Rs 120 and wants to buy some bats and balls. The cost of one register is Rs 40 and that of a pen is Rs 20. In this case, if x denotes the number of bats and y, the number of balls that Ram buys, then the total amount spent by her is Rs (40x + 20y), and we have
40x + 20y ≤ 120
A statement involving the symbols '
(i) Inequalities that do not involve variables are called numerical inequalities. For example,
(ii) Inequalities that involve variables are called literal inequalities. For example,
(iii) An inequality may contain more than one variable and it can be linear, quadratic cubic, etc. For example,
(iv) Inequalities involving the symbol '
(v) Inequalities involving the symbol '
(i) The value(s) of the variable(s) which make the inequality a true statement is called its solutions. The set of all solutions of an inequality is called the solution set of the inequality. For example,
To solve an inequality, we can
(i) Add (or subtract) the same quantity to (from) both sides without changing the sign of inequality.
(ii) Multiply (or divide) both sides by the same positive quantity without changing the sign of inequality. However, if both sides of inequality are multiplied (or divided) by the same negative quantity, the sign of inequality is reversed, i.e., '
To represent the solution of a linear inequality in one variable on a number line, we use the following conventions:
(i) If the inequality involves '
(ii) If the inequality involves '
(a) To represent the solution of a linear inequality in one or two variables graphically in a plane, we proceed as follows:
(i) If the inequality involves '
(ii) If the inequality involves '
(b) Solution of a linear inequality in one variable can be represented on the number line as well as in the plane but the solution of a linear inequality in two variables of the type
(c) Two or more inequalities taken together comprise a system of inequalities, and the solutions of the system of inequalities are the solutions common to all the inequalities comprising the system.
Equal numbers may be subtracted from or added to both sides of an inequality without affecting the inequality sign.
Example:
Solve 5x – 3 < 3x +1
when (i) x is an integer
(ii) x is a real number.
Solution:
We have,
5x –3 < 3x + 1
or 5x –3 + 3 < 3x +1 +3 (Rule 1)
or 5x < 3x +4
or 5x – 3x < 3x + 4 – 3x (Rule 1)
or 2x < 4 or x < 2 (Rule 2)
When x is an integer, the solutions of the given inequality are ...,
– 4, – 3, – 2, – 1, 0, 1
When x is a real number, the solutions of the inequality are given by x < 2, i.e., all real numbers x which are less than 2
Therefore, the solution set of the inequality is x ∈ (– ∞, 2).
We know that a line divides the Cartesian plane into two parts known as half-planes.
The region consisting of all the solutions of an inequality is called the solution region.
If inequality is of the type ax + by ≥ c or ax + by ≤ c, then the points on the line ax + by = c, are also included in the solution region. So draw a dark line in the solution region.
If inequality is of the form ax + by > c or ax + by < c, then the points on the line ax + by = c, are not to be included in the solution region. So draw a broken or dotted line in the solution region.
This is all about this chapter.
Class 11 Linear inequalities notes will help to understand the formulas, statements, and rules in detail. Also, Class 11 Math chapter 6 notes contain previous year’s questions and the NCERT textbook PDF.
After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.
Students can also check these well-structured, subject-wise solutions.
Students should always analyze the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.
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