Linear Inequalities • Linear inequalities in two variables: An
• An inequality is a relation that holds between inequality which involves a linear function in
two values when they are different. two variable E.g. 3x + 2y < 5;
• E.g. x < 5. Here there is a relation between x & 5. • Quadratic inequalities: An inequality which
• 2 real numbers or algebraic expressions related involves a quadratic function .E.g. x2 + 2x ≤ 5
by symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form inequality.
Solution for linear inequality in one variable
• The study of inequalities is very useful in solving Solution & Solution Set
problems in the field of science, mathematics,
• Solution: Values of x, which make inequality a
statistics, optimization problems, economics,
true statement. E.g. 3 is a solution for x<7
psychology, etc.
• Solution Set: The set of values of x is called its
Example of inequality in daily life: solution set. E.g.: {1,2,3,4,5,6} is solution set for
x<7 where x is natural Number
Alex has 200 Rs & wants to buy some pen & eraser.
How many pen & eraser she can buy?
Rules of Inequality:
We can write this statement mathematically using
inequalities. Let the number of pen Alex can buy be • Equal numbers may be added to (or
x & number of eraser be y. Then, 7𝑥 + 23𝑦 ≤ 200. subtracted from) both sides of an inequality
without affecting the sign of inequality. E.g.
Notations 𝑥 < 7 is same as 𝑥 + 2 < 7 + 2
• The notation a < b means that a is less than b. • Both sides of an inequality can be multiplied
• The notation a > b means that a is greater (or divided) by the same positive number
than b. without affecting the sign of inequality.
• The notation a ≠ b means that a is not equal E.g. : 𝑥 + 𝑦 < 7 is same as (𝑥 + 𝑦) × 3 <
to b 7×3
• The notation a ≤ b means that a is less than or • But when both sides are multiplied or
equal to b divided by a negative number, then the sign
• The notation a ≥ b means that a is greater than of inequality is reversed. e.g: 𝑥 + 𝑦 < 7 is
or equal to b same as (𝑥 + 𝑦) × (−3) > 7 × (− 3)
Types of Inequalities: Question: Solve 30 𝑥 < 160 when (i) x is a natural
• Numerical inequalities: Relationship between number, (ii) x is an integer, (iii) x is real number
numbers. E.g. 3 < 5 Solution:
• Literal or variable inequalities: Relationship Dividing the inequality by 30 as per rule 2.
between variables or variable & number. E.g. x 30𝑥 160
<
<5 30 30
16
• Double Inequalities: Relationship from two 𝑂𝑟 𝑥 <
side. E.g. 2 <x < 5 3
Case 1: 𝑥 is a natural number. Then solution set is
• Strict inequalities: An inequality that uses the
{1,2,3,4,5}
symbols < or >. The symbols ≤ and ≥ are not
Case 2: 𝑥 is an integer. Then solution set is {……-4,-
used. g. x < 5; 3<5
3,-32,-1,0,1,2,3,4,5}
• Slack inequalities. An inequality that uses the Case 3: 𝑥 is a real number. Then solution set is
symbols ≤ or ≥ E.g. x ≤ 5 (–∞,16/3).
• Linear inequalities in one variable: An We can represent case 3 solution using number
inequality which involves a linear function in line
one variable E.g. x < 5;
• An inequality is a relation that holds between inequality which involves a linear function in
two values when they are different. two variable E.g. 3x + 2y < 5;
• E.g. x < 5. Here there is a relation between x & 5. • Quadratic inequalities: An inequality which
• 2 real numbers or algebraic expressions related involves a quadratic function .E.g. x2 + 2x ≤ 5
by symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form inequality.
Solution for linear inequality in one variable
• The study of inequalities is very useful in solving Solution & Solution Set
problems in the field of science, mathematics,
• Solution: Values of x, which make inequality a
statistics, optimization problems, economics,
true statement. E.g. 3 is a solution for x<7
psychology, etc.
• Solution Set: The set of values of x is called its
Example of inequality in daily life: solution set. E.g.: {1,2,3,4,5,6} is solution set for
x<7 where x is natural Number
Alex has 200 Rs & wants to buy some pen & eraser.
How many pen & eraser she can buy?
Rules of Inequality:
We can write this statement mathematically using
inequalities. Let the number of pen Alex can buy be • Equal numbers may be added to (or
x & number of eraser be y. Then, 7𝑥 + 23𝑦 ≤ 200. subtracted from) both sides of an inequality
without affecting the sign of inequality. E.g.
Notations 𝑥 < 7 is same as 𝑥 + 2 < 7 + 2
• The notation a < b means that a is less than b. • Both sides of an inequality can be multiplied
• The notation a > b means that a is greater (or divided) by the same positive number
than b. without affecting the sign of inequality.
• The notation a ≠ b means that a is not equal E.g. : 𝑥 + 𝑦 < 7 is same as (𝑥 + 𝑦) × 3 <
to b 7×3
• The notation a ≤ b means that a is less than or • But when both sides are multiplied or
equal to b divided by a negative number, then the sign
• The notation a ≥ b means that a is greater than of inequality is reversed. e.g: 𝑥 + 𝑦 < 7 is
or equal to b same as (𝑥 + 𝑦) × (−3) > 7 × (− 3)
Types of Inequalities: Question: Solve 30 𝑥 < 160 when (i) x is a natural
• Numerical inequalities: Relationship between number, (ii) x is an integer, (iii) x is real number
numbers. E.g. 3 < 5 Solution:
• Literal or variable inequalities: Relationship Dividing the inequality by 30 as per rule 2.
between variables or variable & number. E.g. x 30𝑥 160
<
<5 30 30
16
• Double Inequalities: Relationship from two 𝑂𝑟 𝑥 <
side. E.g. 2 <x < 5 3
Case 1: 𝑥 is a natural number. Then solution set is
• Strict inequalities: An inequality that uses the
{1,2,3,4,5}
symbols < or >. The symbols ≤ and ≥ are not
Case 2: 𝑥 is an integer. Then solution set is {……-4,-
used. g. x < 5; 3<5
3,-32,-1,0,1,2,3,4,5}
• Slack inequalities. An inequality that uses the Case 3: 𝑥 is a real number. Then solution set is
symbols ≤ or ≥ E.g. x ≤ 5 (–∞,16/3).
• Linear inequalities in one variable: An We can represent case 3 solution using number
inequality which involves a linear function in line
one variable E.g. x < 5;
