In the previous section, we completed a discussion on complex numbers and quadratic equations. In this chapter, we will see Linear Inequalities.
In our earlier classes, we have seen statements which can be converted into mathematical equations. Let us see some examples:
Example 1:
• If the cost of one book is Rs 20/-, then the cost of n books will be Rs 20n
• We can write it in the form of an equation:
Total cost = 20n
• A student purchasing books will have to make the payment based on this equation.
Example 2:
• If the cost of one book is Rs 20 and that of a pen is Rs 8, then the cost of x books and y pens will be Rs (20x + 8y)
• We can write it in the form of an equation:
Total cost = 20x + 8y
• A student purchasing books and pens will have to make the payment based on this equation.
• In the above two examples, we effectively converted the statements into mathematical equations ('Equation' indicates 'equality'. Note the ‘=’ sign in both examples).
• But in our day to day life, we will come across some types of statements which are impossible to convert into equations. Let us see some examples:
Example 3:
• If the cost of one book is Rs 20/-, then the cost of n books will be Rs 20n
• If a student has Rs 90/- with him, he will have to make the purchase in such a way that, the total cost is less than 90
• We can write it in a mathematical form:
20n < 90
Example 4:
• If the cost of one book is Rs 20 and that of a pen is Rs 8, then the cost of x books and y pens will be Rs (20x + 8y)
• If a student has Rs 120/- with him, he will have to make the purchase in such a way that, the total cost is less than 120
• We can write it in a mathematical form:
20x + 8y ≤ 120
• In this situation, a doubt will arise in the mind of the reader:
In the third example, the sign is '<'. But in the fourth example, the sign is '≤'. Why is it so?
• The answer can be written in 2 steps:
1. In the third example, we count the number of books using whole numbers 0, 1, 2, 3, 4 etc.,
♦ No whole number when multiplied with 20, will give 90.
♦ In other words, 90 is not a multiple of 20.
♦ So the total cost in this case will never become equal to 90.
♦ Consequently, we cannot use '≤'. We can use only '<'
2. In the fourth example also, we count the number of books and pens using whole numbers.
♦ If the number of books (x) is 4 and number of pens (y) is 5, we get:
♦ Total cost = (20 × 4 + 8 × 5) = (80 + 40) =120
♦ So a total cost equal to 120 is possible.
♦ Consequently, we cannot use '<'. We can use only '≤'
• Consider the four signs given below:
♦ < (less than)
♦ > (greater than)
♦ ≤ (less than or equal)
♦ ≥ (greater than or equal)
• If a statement contains any one of the above four signs, then that statement is called an inequality.
• The statement that we saw in example 3 is: 20n < 90
♦ This statement is an inequality.
• The statement that we saw in example 4 is: 20x + 8y ≤ 120
♦ This statement is also an inequality.
♦ In fact, this statement is a combination of two statements:
✰ (i) 20x +8y < 120
✰ (ii) 20x +8y = 120
♦ Statement (i) is an inequality.
♦ Statement (ii) is an equation.
Definition 1
This definition can be written in 11 steps:
1. If a statement contains any one of the four signs <, >, ≤ or ≥, then that statement is called an inequality.
2. Inequalities can be broadly classified into two categories:
◼ Numerical inequalities
◼ Literal inequalities
• Two real numbers related by <, >, ≤ or ≥ is called a numerical inequality.
For example:
♦ 5 < 8
♦ 11 > 9
• Two algebraic expressions related by <, >, ≤ or ≥ is called a literal inequality.
For example:
♦ x < y
♦ (x+y) > z
• A real number and an algebraic expression related by <, >, ≤ or ≥ is also called a literal inequality.
For example:
♦ x < 8
♦ (x+y) ≥ 2
3. All inequalities that we meet in mathematics, science, engineering, statistics, economics, psychology etc., will fall in one of the above two categories.
4. If an inequality has two of the four signs , it is called a double inequality.
For example:
♦ 2 < 7 < 11
✰ This is read as: 7 is greater than 2 and less than 11
♦ 4 ≤ x < 15
✰ This is read as: x is greater than or equal to 4 and less than 15
♦ 3 < y ≤ 8
✰ This is read as: y is greater than 3 and less than or equal to 8
5. If the inequality involves only < or >, it is called a strict inequality.
For example:
♦ ax+b < 0
♦ ax+b > 0
6. If the inequality involves ≤ or ≥, it is called a slack inequality.
For example:
♦ ax+b ≤ 0
♦ ax+b ≥ 0
7. If the algebraic expression has only one variable, it is called an inequality in one variable.
For example:
♦ ax+b ≤ 0
✰ a and b are constants. x is the variable.
♦ ay+b > c
✰ a, b and c are constants. y is the variable.
8. If the algebraic expression has two variables, it is called an inequality in two variables.
For example:
♦ ax+by < c
✰ a, b and c are constants. x and y are the variables.
♦ ax+by ≥ c
✰ a, b and c are constants. x and y are the variables.
9. If in an "inequality in one variable", the highest exponent of the variable is '1', it is called a linear inequality in one variable.
For example:
♦ ax+b ≤ 0
✰ x is the variable. It's highest exponent is 1
♦ ay+b > c
✰ y is the variable. It's highest exponent is 1
(Recall that if the highest exponent is '1', the graph will be a line. That is why, it is called 'linear')
10. If in an "inequality in two variables", the highest exponent of both variables is '1', it is called a linear inequality in two variables.
For example:
♦ ax+by ≤ 0
✰ x and y are the variables.Highest exponent is 1 for both of them.
♦ ay+by > c
✰ x and y are the variables.Highest exponent is 1 for both of them.
11. If in an "inequality in one variable", the highest exponent of the variable is '2', it is called a quadratic inequality in one variable. It's graph will not be a line.
For example:
♦ ax2+bx + c ≤ 0
✰ x is the variable. It's highest exponent is 2
♦ ax2+bx > c
✰ x is the variable. It's highest exponent is 2
In this chapter, we will learn about inequalities that fall in the following two categories:
(i) Linear inequality in one variable.
(ii) Linear inequality in two variables.
Algebraic solutions of linear inequalities in one variable
• Consider the inequality (20n < 90) that we saw in example 3 above. An analysis about this inequality, can be written in 5 steps:
1. We want to find the value of n (the number of books).
• If the correct value of n can be calculated, the cost of purchase will not go above Rs 90/-
• n cannot be a -ve number because, we cannot buy -ve number of books.
• Also n cannot be decimals or fractions like 2.5, 3/2 etc., because, a book cannot be made into fractions.
2. So it is obvious that, n must be a whole number. Let us try various whole numbers:
(i) Put n = 0,
♦ the left hand side (LHS) become (20 × 0) = 0
♦ the right hand side (RHS) is always 90
♦ 0 < 90
`• So when n = 0, the statement is true.
(ii) Put n = 1,
♦ LHS become (20 × 1) = 20
♦ RHS is always 90
♦ 20 < 90
`• So when n = 1, the statement is true.
(iii) Put n = 2,
♦ LHS become (20 × 2) = 40
♦ RHS is always 90
♦ 40 < 90
`• So when n = 2, the statement is true.
(iv) Put n = 3,
♦ LHS become (20 × 3) = 60
♦ RHS is always 90
♦ 60 < 90
`• So when n = 3, the statement is true.
(v) Put n = 4,
♦ LHS become (20 × 4) = 80
♦ RHS is always 90
♦ 80 < 90
`• So when n = 4, the statement is true.
(vi) Put n = 5,
♦ LHS become (20 × 5) = 100
♦ RHS is always 90
♦ 100 ≮ 90
`• So when n = 5, the statement is false.
3. We can write:
The values of n which make the inequality true, are: 0, 1, 2, 3, 4
4. Now we can define solutions:
◼ The values which make the inequality true are called solutions of inequality.
5. Also we can define solution set:
◼ The set containing all the solutions of an inequality is called the solution set of that inequality.
◼ Such a set must contain only the solutions. Numbers which are not solutions must not be included in the solution set.
• So in our present case, the solution set is {0, 1, 2, 3, 4}
Let us see the analysis of another example. It can be written in 7 steps:
1. A car parking area can accommodate a maximum of 15 cars. There are 9 cars already parked. What is the additional number of cars that can be allowed?
2. We can write the statement as an inequality: 9+x ≤ 15
Where x is the possible number of additional cars.
3. We want to find the value of x (the number of cars).
• If the correct value of x can be calculated, the parking will be OK.
• x cannot be a -ve number because, number of cars cannot be -ve.
• Also x cannot be decimals or fractions like 2.5, 3/2 etc., because, a car cannot be made into fractions.
4. So it is obvious that, x must be a whole number. Let us try various whole numbers:
(i) Put x = 0,
♦ the left hand side (LHS) become (9+0) = 9
♦ the right hand side (RHS) is always 15
♦ 9 ≤ 15
`• So when x = 0, the statement is true.
(ii) Put x = 1,
♦ LHS become (9+1) = 10
♦ RHS is always 15
♦ 10 ≤ 15
`• So when x = 1, the statement is true.
(iii) Put x = 2,
♦ LHS become (9+2) = 11
♦ RHS is always 15
♦ 11 ≤ 15
`• So when x = 2, the statement is true.
(iv) Put x = 3,
♦ LHS become (9+3) = 12
♦ RHS is always 15
♦ 12 ≤ 15
`• So when x = 3, the statement is true.
(v) Put x = 4,
♦ LHS become (9+4) = 13
♦ RHS is always 15
♦ 13 ≤ 15
`• So when x = 4, the statement is true.
(vi) Put x = 5,
♦ LHS become (9+5) = 14
♦ RHS is always 15
♦ 14 ≤ 15
`• So when x = 5, the statement is true.
(vii) Put x = 6,
♦ LHS become (9+6) = 15
♦ RHS is always 15
♦ 15 ≤ 15
`• So when x = 6, the statement is true.
(viii) Put x = 7,
♦ LHS become (9+7) = 16
♦ RHS is always 15
♦ 16 ≰ 15
`• So when x = 7, the statement is false.
5. We can write:
The values of x which make the inequality true, are: 0, 1, 2, 3, 4, 5, 6
6. Now we can define solutions:
◼ The values which make the inequality true are called solutions of inequality.
7. Also we can define solution set:
◼ The set containing all the solutions of an inequality is called the solution set of that inequality.
◼ Such a set must contain only the solutions. Numbers which are not solutions must not be included in the solution set.
• So in our present case, the solution set is {0, 1, 2, 3, 4, 5, 6}
• We have seen two examples. In both those examples, we found out the
solution set using trial and error method.
• But the trial and error
method is time consuming. Also it may not work in all cases.
• So we must
develop a more systematic method. We will see such a method in the next section.
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