**Write a Linear Inequality Word Problems :**

To write a linear inequality from word problems, first we have to understand the information given in the question clearly.

Use an inequality to model and solve the following problems

**Example 1 :**

Find all pairs of consecutive odd positive integers, both of which are smaller than 18, such that their sum is more than 20.

**Solution :**

Let "x" be the smaller of the two consecutive odd positive integers. Then, the other odd integer is "x+2"

It is given that both the integers are smaller than 18 and their sum is more than 20.

Hence, x + 2 < 18 -------(1)

and x + (x + 2) > 20 -------(2)

From (1)

x + 2 < 18

Subtract 2 on both sides

x + 2 - 2 < 18- 2

x < 16

From (2)

x + (x + 2) > 20

2x + 2 > 20

Subtract 2 on both sides

2x + 2 - 2 > 20- 2

2x > 18

Divide by 2 on both sides

2x/2 > 18/2

x > 9

From this we come to know that the required two numbers are between 9 and 16.We should not take 9 and 16.

x = 11, 13, 15

Hence, the required pairs of odd integers are

(11, 13) ,(13, 15) and (15, 17)

**Example 2 :**

Find all pairs of consecutive even positive integers, both of which are larger than 8, such that their sum is less than 25.

**Solution :**

Let "x" be the smaller of the two consecutive even positive integers. Then, the other odd integer is "x + 2"

It is given that both the integer are larger than 8 and their sum is less than 25.

So,

x > 8 -------(1)

x + x + 2 < 25 --------(2)

From (1)

x > 8

Form (2)

2 x + 2 < 25

Subtract 2 on both sides

2x + 2 - 2 < 25 - 2

2x < 23

Divide by 2 on both sides, we get

2x/2 < 23/2

x < 23/2

From this we come to know that the required numbers are between 8 and 12.5.We should not take 8 and 12.5.

x = 10

Hence the required pair is

(10, 12)

**Example 3 :**

In the first four papers each of 100 marks, John got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks, find the range of marks he should score in the fifth paper.

**Solution :**

John scores x marks in the fifth paper. Then,

75 ≤ [(95 + 72 + 73 + 83 + x)/5] < 80

75 ≤ (323 + x)/5 < 80

Multiply by 5

75(5) ≤ (323 + x) < 80(5)

375 ≤ (323 + x) < 400

Subtract 323

375 - 323 ≤ (323 + x - 323) < 400 - 323

52 ≤ x < 77

Hence, John must score between 52 and 77 marks.

**Example 4 :**

A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and third length is to be twice as the shortest. What are the possible lengths for the shortest board if third piece is to be at least 5 cm longer than the second ?

**Solution :**

Let the length of the shortest piece be x cm, Then the lengths of the second and third piece are x + 3 and 2x cm respectively.

x + (x + 3) + 2x ≤ 91 -----(1)

and 2x ≥ (x + 3) + 5 -----(2)

From (1)

2x + 3 + 2x ≤ 91

4x + 3 ≤ 91

Subtract 3 on both sides

4x + 3 - 3 ≤ 91 - 3

4x ≤ 88

Divide by 4 on both sides

(4x)/4 ≤ 88/4

x ≤ 22

From (2)

2x ≥ (x + 3) + 5

2x ≥ x + 8

Subtract x on both ides

2x - x ≥ x - x + 8

x ≥ 8

8 ≤ x ≤ 22

Hence, the shortest piece must be at least 8 cm long but not more than 22 cm long.

After having gone through the stuff given above, we hope that the students would have understood, how to write a linear inequality word problem.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**