Solving Inequalities and Graphing

Definition of inequalities

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 following types,

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


Explanation on types of inequalities:

1) Numerical inequalities:

Inequalities, which involves numerical only without any variables are named numerical inequalities. The example of numerical inequalities is following as,

Eg: 2< 6; 5 >1

2) Literal inequalities:

Inequalities, which involve one or more variables, are named literal inequalities.

Eg: a< 5; b >2; x ≥4; y≤ 6

3) Double inequalities:

An inequality which have two signs (< or > or ≤ or ≥) is described double inequality.

Eg: 2< b< 8; 2 ≥y ≥5

4) Strict inequalities:

If an inequality contains a sign < or >, then it is referred strict inequalities

Eg: Ax + B< 0; Ax2 + Bx + C >0

5) Slack inequalities:

If an inequality involves a sign ≤ or ≥, then it is referred as slack inequalities.

Eg: Ax + By≤ C; Ax + By ≥C

6) Linear inequalities:

An inequality may involve one variable by linear is labeled as linear inequality with one variable; If it involves two variables, then it is labeled linear inequality with two variables.

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

Examples on inequalities:

1) Solving and graphing the following inequality: 20v < 200 when

(i) ‘v’ is a natural number,

(ii) ‘v’ is an integer.

Solution:

Solving 20 v < 200

For solving the inequality,

20v/ 20< 200 / 20

v< 10.

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

1, 2, 3, 4, 5, 6, 7, 8, 9

The solution set is {1, 2, 3, 4, 5, 6, 7, 8, and 9}.

Then graphing of this function is,

Solution of the graph v < 10 (V is natural)



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

..., – 3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

The solution set is {...,–3, –2,–1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Graphing the above solution set as

Solution of the graph v < 10 (V is integer)

2) Solving and graphing the following inequality: 2x + 3< 8x +9.

Solution:

For solving and graphing the inequalities,

2x + 3 < 8x + 9

Thus, basic rules are used for solving and graphing the inequality,

2x +3 – 3 < 8x + 9 – 3

2x < 8x + 6

2x – 8x < 8x + 6 – 8x

- 6x < 6

x > – 1

The solution set is {0, 1, 2, 3…}

Graphing of above solution set is,

Solution of the graph x> -1