Inequality Exam Preparation

Introduction to inequality exam preparation:

Inequality is defined as two real numbers or two algebraic expressions are related with functioning a sign as ‘<’ (less than), ‘>’ (greater than), ‘≤’ (less than or equal) and ≥ (greater than or equal). Inequalities are classified as follows,

Numerical inequalities
Literal inequalities
Double inequalities
Strict inequalities
Slack inequalities
Linear inequalities

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Inequality exam preparation - Types of Inequality:


1) Numerical inequalities:

Inequalities which contain numerical only without any variables are called numerical inequalities.

Eg: 2< 6; 5 >1

2) Literal inequalities:

Inequalities which contain one or more variables are called literal inequalities.

Eg: a< 5; b >2; x ≥4; y≤ 6

3) Double inequalities:

An inequality which involves two symbol (< or > or ≤ or ≥) is called double inequality.

Eg: 2< 6< 8; 2 ≥y ≥5

4) Strict inequalities:

If an inequality includes a symbol < or >, then it is called strict inequalities.

Eg: Ax + B< 0; Ax^2 + Bx + C >0

5) Slack inequalities:

If an inequality includes a symbol ≤ or ≥, then it is called slack inequalities.

Eg: Ax + By≤ C; Ax + By ≥C

6) Linear inequalities:

An inequality may include one variable with linear is called linear inequality with one variable; If it contains two variables, then it is called linear inequality with two variables.

Eg: Ax + By< C; Ax + B >C

Inequality exam preparation - Rules:

1: Both sides of an inequality can be added to or subtracted by equal numbers without affecting the sign.

2: Equal numbers may be multiplied or divided from both sides of an inequality.

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Inequality exam preparation - Examples:


Example 1:

Solve 40a < 200 when

(i) ‘a’ is a natural number,

(ii) ‘a’ is an integer.

Solution:

Given 40 a < 200

40a/ 40< 200 / 40 (Rule 2)

a< 5.

(i) When ‘a’ is a natural number, then the statement gives,

1, 2, 3, 4.

The solution set is {1, 2, 3, and 4}.

(ii) When ‘a’ is an integer, then the solution is given as,

..., – 3, –2, –1, 0, 1, 2, 3, 4.

The solution set is {...,–3, –2,–1, 0, 1, 2, 3, 4}

Example 2:

Solve 4x + 3< 8x +7.

Solution:

4x + 3 < 8x + 7

4x +3 – 3 < 8x + 7 – 3 (Rule 1)

4x < 8x + 4

4x – 8x < 8x + 4 – 8x

- 4x < 4 (Rule 2)

x > – 1

The solution set is {0, 1, 2, 3…}