**Solving Linear Inequalities in One Variable Word Problems :**

In this section, we will learn, how to solve word problems using linear inequalities.

To solve word problems using linear inequalities, we have to model the information given in the question as linear inequalities and solve for unknown.

**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).

**Given : **Both the integers are smaller than 18.

In solving this kind of problems, when both the smaller and larger integers are less than 18, always we have to take the larger integer to form inequality.

Then, we have

x + 2 < 18

Subtract 2 from each side.

x < 16 -----(1)

**Given : **Sum of the integers is more than 20.

Then, we have

x + (x + 2) > 20

x + x + 2 > 20

2x + 2 > 20

Subtract 2 from each side.

2x > 18

Divide each side by 2.

x > 9

9 < x -----(2)

Combine (1) and (2).

9 < x < 16

So, the value of x is any odd integer between 9 and 16.

The possible values of x are

11, 13, 15

When x = 11, 12, 13, the possibles values of (x + 2) are

13, 15, 17

Therefore, 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 odd positive integers.

Then, the other odd integer is (x + 2).

**Given : **Both the integers are larger than 8.

In solving this kind of problems, when both the smaller and larger integers are larger than 18, always we have to take the smaller integer to form inequality.

Then, we have

x > 8

8 < x -----(1)

**Given : **Sum of the integers is less than 25.

Then, we have

x + (x + 2) < 25

x + x + 2 < 25

2x + 2 < 25

Subtract 2 from each side.

2x < 23

Divide each side by 2.

x < 11.5 -----(2)

Combine (1) and (2).

8 < x < 11.5

So, the value of x is any even integer between 8 and 11.5.

There is only one even integer between 8 and 11.5

So, the possible value of x is

10

When x = 10, the possibles value of (x + 2) is

12

Therefore, the required pair of even integers 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 :**

Let x be the marks scored by John in fifth paper.

Then, we have

75 ≤ [(95 + 72 + 73 + 83 + x)/5] < 80

Simplify.

75 ≤ (323 + x) / 5 < 80

Multiply each side by 5.

375 ≤ (323 + x) < 400

Subtract 323 from each side.

52 ≤ x < 77

So, John should score between 51 and 77 marks in his fifth paper.

**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 piece, if third piece is to be at least 5 cm longer than the second ?

**Solution :**

Let x be the length of the shortest piece

Then, the lengths of the second and third piece are

x + 3 and 2x

**Given :** The total length of the board is 91 cm.

Then, we have

x + (x + 3) + 2x ≤ 91

Simplify.

4x + 3 ≤ 91

Subtract 3 from each side.

4x ≤ 88

x ≤ 22 -----(1)

**Given :** The third piece has to be at least 5 cm longer than the second.

Then, we have

2x ≥ (x + 3) + 5

2x ≥ x + 8

Subtract x from each side.

x ≥ 8

8 ≤ x -----(2)

Combine (1) and (2).

8 ≤ x ≤ 22

So, 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 solve word problems on linear inequalities in one variable.

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**

**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**