CHSE Odisha Class 11 Math Notes Chapter 7 Linear Inequalities

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Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 7 Linear Inequalities will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 7 Linear Inequalities

Inequality:
A statement with symbols like >, ≥, <, ≤ is an inequality.

Different types of inequality:

(a) Numerical inequality: It is an inequality involving numbers not variables.
(b) Literal inequality: It is the inequality involving literal numbers(variable).
(c) Strict inequality: An inequality with only > or < symbols is a strict inequality.
(d) Slack inequality: An inequality with only ≥ or ≤ symbols is a slack inequality.

Linear inequality:
An inequality involving variables in the first degree is called linear inequalities.
(a) General form of inequalities:
(i) In one variable: ax + b > or ≥ or < or ≤ 0
(ii) In two variables: ax + by + c > or ≥ or < or ≤ 0.

Intervals:

  • Closed Interval: [a, b] = {x ∈ R: a ≤ x ≤ b}
  • Open Interval: (a, b) = {x ∈ R: a < x < b}
  • Semi-open or semi-closed interval:
    ⇒ [a, b) = {x ∈ R: a ≤ x < b}
    ⇒ (a, b] = {x ∈ R: a < x ≤ b}

Basic properties of inequalities:
(1) a > b, b > c ⇒ a > c
(2) a > b ⇒ a ± c > b ± c
(3) a > b

  • m > 0 ⇒ am > bm, \(\frac{a}{m}>\frac{b}{m}\)
  • m < 0 ⇒ am < bm, \(\frac{a}{m}<\frac{b}{m}\)

(4) If a > b > 0, then
a2 > b2, |a| > |b| and \(\frac{1}{a}>\frac{1}{b}\)
If a < b < 0, then
|a| > |b| and \(\frac{1}{a}>\frac{1}{b}\)

CHSE Odisha Class 11 Math Notes Chapter 7 Linear Inequalities

Graphical solution of linear inequalities in two variables:
Working rule:

Let the inequality is ax + by + c < or ≤ or > or ≥ 0

Step – 1: Consider the equation ax + by + c = 0 in place of the inequality and draw its graph (Draw a dotted line for > or < and a bold line for ≥ or ≤).
Step – 2: Take any point that does not lie on the graph, and put the coordinate in the inequality.
If you get true then the inequality is satisfied. Shade the half-plane containing that point otherwise the inequality is not satisfied. In this case shade the half plane region that does not contain the point.
Step – 3: The shaded region is the required solution.

Solution of a system of linear inequalities in two variables:

Step – 1: Draw the graph of all lines.
Step – 2: Shade the appropriate region for each inequality.
Step – 3: The common region is the required solution.

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