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Linear Inequalities
Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form
an inequality
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Example: 6 < 26, 3 < z + 1 ≤ 22, 27 ≥ s ≥ 16, p + t > 100
Example 1:
A shopkeeper sells some television sets and A/C sets every day
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For this, he must sell 20 television sets and 30 A/C sets every
week or he must sell 50 television sets and 10A/C sets every week
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Solution:
Let the profit on each television set be Rs x and the profit on each A/C be Rs y
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Hence, the total profit earned by the shopkeeper by selling 20
television sets and 30 A/C sets is Rs (20x + 30y)
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Hence, the total profit earned by the
shopkeeper by selling 50 television sets and 10 A/C sets is Rs (50x + 10y)
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Hence, the given situation can be expressed mathematically by the following two inequalities:
20x + 30y ≥ 10000 and 50x + 10y ≥ 10000

Any solution of an inequality in one variable is a value of the variable that makes it a true
statement
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The rules that need to be followed to solve an inequality are:
Equal numbers may be added to (or subtracted from) both sides of an inequality
without affecting the sign of the inequality
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However, when both sides are multiplied or divided by a negative
number, then the sign of the inequality is reversed
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Example:
Show the graph of the solution of the inequality 5(x – 3) > 2x + 9 on number line
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The solution set might be taken from real numbers or whole numbers or integers or any
other set of numbers
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We may take any set as the
replacement set
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Linear inequalities in two variables and representing their solution graphically
Rules for solving an inequality:
Equal numbers may be added to or subtracted from both sides of an inequality
without affecting the sign of the inequality
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But when both sides are multiplied with or divided by a negative number, the
sign of inequality is reversed
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Solution:

⇒ 3(3x + 20) ≥ 10(x – 3)
⇒ 9x + 60 ≥ 10x – 30
⇒ 9x – 10x ≥ – 30 – 60
⇒ – x ≥ – 90
⇒ x ≤ 90
∴ The solution set of the given inequality is (– ∞, 90]
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In order to identify the half plane represented by an inequality, it is sufficient to take any point
(a, b) (not on the line) and check whether it satisfies the inequality or not
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If not, then
the inequality represents the half plane which does not contain the point
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In an inequality of the type ax + by ≥ c or ax + by ≤ c, the points on the line ax + by = c are to be
included in the solution region
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In an inequality of the type ax + by > c or ax + by < c, the points on the line ax + by = c are not
to be included in the solution region
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Example 2: Solve the following system of linear inequalities graphically: 2x – y ≤ 4, x < 2
Solution:The given linear inequalities are
2x – y ≤ 4
… (i)
x<2
… (ii)
The graphs of the lines 2x – y = 4 and x = 2 are drawn in the figure below
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Inequality (ii) represents the region on the left of the line x = 2 (excluding the line x = 2)
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