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Linear Inequation

Linear inequality is an inequality which involves a linear function and it contains one of the symbols of inequality.

  • < is less than
  • > is greater than
  • \leq  is less than or equal to
  • \geq  is greater than or equal to
  • \neq  is not equal to

A linear inequality actually looks exactly like a linear equation   (y = mx + c) , with the inequality sign replacing the equality sign   (e.g. y > mx + c).

Two-dimensional linear inequalities are expressions in 2 variables of the form:  ax + by < c \ \ or\ \  ax + by \le c   where the inequalities may either be strict or not.

A statement of any of the following forms:    i) ax + b > 0  \ \ ii) ax + b < 0   \ \ iii) ax + b \ge 0       \ \ or  \ \ iv) ax + b \le 0   where  a    and   b   are real numbers and  is called a Linear Inequation in x .

m46xReplacement Set or Universal Set: This is a set which contains all the values of the variable  x  , which would satisfy the inequation.

Solution Set: This is a subset of the replacement set which satisfy the given inequation.

 

Properties of Inequalities

Addition and Subtraction Property

A common constant c may be added to or subtracted from both sides of an inequality. For any real numbers

If \ a \le b, \ then \ a + c \le b + c \ and \ a - c \le b - c

 If \ a \ge b, \ then \ a+b \ge b+c \ and \ a-c \ge b-c  

Multiplication and Division Property. The property states that for any real numbers,  a, b \ and \ non-zero \  c:

If  \  c \ is \  positive  , then multiplying or dividing by c does not change the inequality:

 If a \ge b \ and \  c > 0, \ then \  ac \ge bc \ and\   \frac{a}{c}  \ge\frac{b}{c}

 If \  a \le b \ and \ c > 0, \ then \  ac \le bc \  and \   \frac{a}{c} \le\frac{b}{c}

If     \ c  \  \ is \  \  negative  , then multiplying or dividing by  c   inverts the inequality:

If a \ge b \ and \ c < 0, \ then \ ac \le bc \ and \ \frac{a}{c}  \le  \frac{b}{c}  

If  \ a \le b \ and \ c < 0, \ then \ ac \ge bc \ and \ \frac{a}{c} \ge \frac{b}{c}  

Transitive property of inequality states that for any real numbers  \ a, \ b, \ c   :

If  \ a \ge b \  and  \ b \ge c, \ then  \ a \ge c 

If \  a \le b  \ and  \ b \le c, \ then  \ a \le c 

Converse Property: The relations ≤ and ≥ are each other’s converse. For any real numbers   \ a  \ and \  b:

 If  \ a \le b, \ then  \ b \ge a

 If  \ a \ge b, \ then \  b \le a

Additive inverse: The properties for the additive inverse state: For any real numbers      \ a  \ and \  b  , negation inverts the inequality:

 If a \le b, then -a \ge -b

 If a \ge b, then -a \le -b

m47xMultiplicative Inverse: The properties for the multiplicative inverse state:

For any non-zero real numbers   \ a  \ and \  b   that are both positive or both negative:

 If a\le b, \ then \ \frac{1}{a} \ge \frac{1}{b}

 If a\ge b, \ then \ \frac{1}{a} \le \frac{1}{b}

If anyone of   \ a  \ and \  b   is positive and the other is negative, then:

 If a< b, \ then \ \frac{1}{a} < \frac{1}{b}

 If a> b, \ then \ \frac{1}{a} > \frac{1}{b}

For any non-zero real numbers   \ a \  and \  b  :

 If 0 < a\le b,  \ then \ \frac{1}{a} \ge \frac{1}{b} > 0

 If a \le b < 0, \ then \ 0 >\frac{1}{a} \ge \frac{1}{b}  

 If a< 0 < b, \ then \ \frac{1}{a} < 0 <\frac{1}{b}

 If 0 > a \ge b, \ then \ \frac{1}{a}  \le \frac{1}{b} < 0

 If a \ge b > 0, \ then \ 0 <\frac{1}{a} \le \frac{1}{b}

 If a> 0 > b, \ then \ \frac{1}{a} > 0 >\frac{1}{b}

 

ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
Class 8: Reference Books               Class 8: NTSE Preparation
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