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Linear Inequations 1

The document explains linear inequations, defining them as mathematical statements where one quantity is not equal to another. It outlines the representation of linear inequalities in one variable, rules for solving them algebraically, and the concepts of replacement and solution sets. Additionally, it describes how to represent solutions on a number line and combine inequalities.

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Piupa Banerjee
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30 views3 pages

Linear Inequations 1

The document explains linear inequations, defining them as mathematical statements where one quantity is not equal to another. It outlines the representation of linear inequalities in one variable, rules for solving them algebraically, and the concepts of replacement and solution sets. Additionally, it describes how to represent solutions on a number line and combine inequalities.

Uploaded by

Piupa Banerjee
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Linear Inequations

Inequation:
The mathematical statement in which the quantity on one side is not equal to the quantity on the
other side is called an inequation.
Representation of linear inequality in one variable:
Let a, b and c be real numbers.

ax + b > c ax + b is greater than c

ax + b < c ax + b is less than c

ax + b ≥ c ax + b is greater than or equal to c

ax + b ≤ c ax + b is less than or equal to c

The signs '>'; '<'; '≤'; '≥' are called signs of inequality.
Solving a Linear Inequality Algebraically:

When a positive term is moved from one side of


Rule 1 an inequality to another, the sign of the term 2x + 3 > 7 = 2x > 7 – 3
becomes negative

When a negative term is moved from one side of


Rule 2 an inequality to another, the sign of the term 2x – 3 > 7 = 2x > 7 + 3
becomes positive

1
When each term of an inequality is multiplied or
x < y = px <py
Rule 3 divided by the same positive number (p), the sign
x < y = x/p < y/p
of the inequality remains unchanged

When each term of an inequality is multiplied or


x < y = px >py
Rule 4 divided by the same negative number (p), the
x < y = x/p > y/p
sign of the inequality reverses

If the sign of each term on both the sides of an


Rule 5 inequality is changed, the sign of inequality gets –x>5=x<–5
reversed

If both the sides of an inequality are either


Rule 6 positive or negative, then on taking their x ≥ y = 1/x ≤ 1/y
reciprocals, the sign of inequality reverses

Replacement Set and Solution Set:


 The set, from which the value of the variable x is to be chosen, is called the replacement
set.
 The subset of the replacement set, whose elements satisfy the given inequality is called
the solution set.
Let the given inequality be x < 3 then:
1. If the replacement set = N, the set of natural numbers, then the solution set = {1, 2}
2. If the replacement set = Z or I, the set of integers, then the solution set = {2, 1, 0, −1, −2}
3. If the replacement set = R, the set of real numbers, then the solution set is {x: x ϵϵ R and x
< 3}
Representation of the solution on the number line
Convention
 A darkened circle on a number indicates that the number is also included in the solution
set. (i.e. ≤ or ≥)
 A hollow circle on a number indicates that the number is not included in the solution set.
(i.e. > or <)
2
Combining Inequalities:
 Simplify the given inequality.
 Plot the solution set of each inequality on the number line.
 Study the solution sets and find out common points or regions of the given inequalities.

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