# Linear Inequations

In general a **linear inequation** can always be written as

a x +b < 0, a x +b 0, a x +b > 0 or a x +b 0,

where a and b are real numbers, a 0.

### Replacement set

The set from which values of the variable (involved in the inequation) are chosen is called the replacement set.

### Solution set

A solution to an inequation is a number (chosen from replacement set) which, when substituted for the variable, makes the inequation true. The set of all solutions of an inequation is called the solution set of the inequation.

For example, consider the inequation x < 4

Replacement set Solution set

(i) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3}

(ii) {-1, 0, 1, 2, 5, 8} {-1, 0, 1, 2}

(iii) {-5, 10} {-5}

(iv) {5, 6, 7, 8, 9, 10}

Note that the solution set depends upon the replacement set.

### Remark

Always reverse the symbol of an inequation when multiplying or dividing by a negative number.

## Illustrative Examples

### Example

Find the solution set of 2 ;< 3 (x - 2) +5 < 8, x
**W.** Also represent its solution on the number line.

### Solution

Given 2 3 (x -2) +5 < 8

=> 2 3 x -6 + 5 < 8

=> 2 3 x -1 < 8

=> 2 +1 3 x -1 +1 < 8 +1
[Add 1]

=> 3 3 x < 9

=> 1 x < 3 [Divide by 3]

But x **W** i.e. x
{0, 1, 2, 3 , 4, ...},

Hence the solution set is {1, 2}

The solution set is shown by thick dots on the number line.

### Example

John needs a minimum of 360 marks in four tests in a Mathematics course to obtain an A grade. On his first three tests, he scored 88, 96, 79 marks. What should his score be in the fourth test so that he can make an A grade?

### Solution

Let John score x marks in the fourth test. Then the sum of Johns test
scores should be greater than or equal to 360 i.e.

88 +96 +79 +x 360 => 263 + x 360

=> x 360 +(-263) => x
97

John should score 97 or greater than 97 in the fourth test to obtain A grade.

## Exercise

- Solve the inequation 3 x -11 < 3 where x {1, 2, 3, ..., 10}.
- Solve 2 (x -3) < 1, x {1, 2, 3, ..., 10}.
- Solve 5 -4 x > 2 -3x, x W.
- List the solution set of 30 -4(2x -1) < 30, given that x is a positive integer.
- Solve 2 (x -2) < 3x -2, x {-3, -2, -1, 0, 1, 2, 3}.
- If x is a negative integer, find the solution set of

2/3 + (x +1)/3 > 0. - Solve (2 x -3)/4, x { 0, 1, 2, ..., 8 }.
- Solve x -3 (2 +x) > 2 (3 x -1), x {-3, -2, -1, 0, 1, 2 }.
- Given x {1, 2, 3, 4, 5, 6, 7, 9}, solve x -3 < 2x -1.
- Given A = {x ; x I, -4 x < 4}, solve 2x - 3 < 3 where x has the domain A.
- List the solution set of the inequation

(1/2) +8 x > 5x - 3/2, x Z - List the solution set of (11 -2x)/5 (9 -3x)/8 + 3/4, x N
- Find the values of x, which satisfy the inequation

-2 1/2 - 2 x/3 11/6, x N - If x W, find the solution set of

3 x/5 - (2 x -1)/3 > 1

Also graph the solution set on the number line, if possible. - Given x {1, 2, 3, 4, 5, 6, 7, 9}, find the values of x for which -3 < 2 x -1 < x +4.
- Solve 1 15 -7x > 2x -27, x N.
- If x Z, solve 2 +4x < 2x -5 < 3x.
- Solve the inequality 2 x -10 < 3 x -15.
- Find the solution set of the inequation x +5 = 2 x + 3, x R.

## Answers

**1.**{1, 2, 3,4}

**2.**{1, 2, 3}

**3.**{0, 1, 2}

**4.**{1, 2, 3, 4, ...}

**5.**{-1, 0, 1, 2, 3}

**6.**{-2, -1}

**7.**{3, 4, 5, 6, 7, 8}

**8.**{-3, -2, -1}

**9.**{1, 2, 3, 4, 5, 6, 7, 9}

**10.**{-4, -3, -2, -1, 0, 1, 2}

**11.**{0, 1, 2, 3, ...}

**12.**{1, 2, 3, ..., 13}

**13.**{1, 2, 3}

**14.**; not possible

**15.**{1, 2, 3, 4}

**16.**{2, 3, 4}

**17.**{-5, -4}

**18.**{x ; x R, x > 5}

**19.**{x ; x R, x 2}