NCERT Class 11-Math's: Chapter –6 Linear Inequalities Part 1

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6.1 Overview

A statement involving the symbols is called an inequality.

For example .

(i) Inequalities which do not involve variables are called numerical inequalities.

For example .

(ii) Inequalities which involve variables are called literal inequalities.

For example, .

(iii) An inequality may contain more than one variable and it can be linear, quadratic or cubic etc. For example, is a linear inequality in one variable, is a linear inequality in two variables and is a quadratic inequality in one variable.

(iv) Inequalities involving the symbol or are called strict inequalities.

For example, .

(v) Inequalities involving the symbol or are called slack inequalities.

For example, .

6.1.2 Solution of an inequality

(i) The value(s) of the variable(s) which makes 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, , has infinite number of solutions as all real values greater than or equal to one make it a true statement. The inequality has no solution in R as no real value of x makes it a true statement.

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., changes into and vice versa.

6.1.3 Representation of solution of linear inequality in one variable on a number line

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 or , we draw filled circle on the number line to indicate that the number corresponding to the filled circle is included in the solution set.

(ii) If the inequality involves or , we draw an open circle on the number line to indicate that the number corresponding to the open circle is excluded from the solution set.

6.1.4 Graphical representation of the solution of a linear inequality

(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 or , we draw the graph of the line as a thick line to indicate that the points on this line are included in the solution set.

(ii) If the inequality involves or , we draw the graph of the line as dotted line to indicate that the points on the line are excluded from the solution set.

(b) Solution of a linear inequality in one variable can be represented on number line as well as in the plane but the solution of a linear inequality in two variables of the type can be represented in the plane only.

(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.

6.1.5 Two important results

(a) If , and , then

(i) or and are of the same sign.

(ii) or and are of opposite sign.

(b) If a is any positive real number, i.e., , then

(i)

(ii)

6.2 Solved Examples

Short Answer Type

Question 1:

Solve the inequality, , when

(i) is a natural number

(ii) is a whole number

(iii) is an integer

(iv) is a real number

Answer:

We have

(Adding 5 to both sides)

(Subtracting x from both sides)

(Dividing by on both sides)

(i) Solution set is

(ii) Solution set is

(iii) Solution set is

(iv) Solution set is , i.e., any real number less than .

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