**Linear Inequalities Word Problems :**

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

**Step 1 :**

Read and understand the information carefully and translate the statements into linear inequalities.

**Step 2 :**

Solve for the variable using basic operations like addition, subtraction, multiplication and division.

**Step 3 :**

Find the solution set and obtain some of the possible solutions.

Apart from the above steps, we have to make the following changes, when we multiply or divide each side of the inequality by a negative value.

- If we have <, then change it as >
- If we have >, then change it as <
- If we have ≤, then change it as ≥
- If we have ≥, then change it as ≤

**Example 1 :**

In 2010 Sacramento, California, received 23 inches in annual precipitation. In 2011, the city received 17 inches in annual precipitation. In which year was there more precipitation ?

**Solution : **

Locate the two integers 23 and 17 on a number line and mark them.

23 is to the right of 17 on the number line.

This means that 23 is greater than 17.

We can write the above situation in terms of inequality as

23 > 17

17 is to the left of 23 on the number line.

This means that 17 is less than 23.

We can write the above situation in terms of inequality as

17 < 23

So, there was more precipitation in 2010.

**Example 2 : **

An employer recruits experienced (x) and fresh workmen (y) for his firm under the condition that he cannot employ more then 9 people. Find the inequality which can relate x and y.

**Solution :**

**Given :** x and y stand for number of experienced person and fresh workmen respectively.

Total number of people recruited is

= x + y

As per the question, total number of people (experienced + fresh) recruited should not be more than 9.

That is, total number of people (x + y) recruited should be equal to 9 or less than 9.

That is

x + y ≤ 9

**Example 3 : **

On the average experienced person does 5 units of work while a fresh one (y) does 3 units of work daily. But the employer has to maintain an output of at least 30 units of work per day. This situation can be expressed as

**Solution :**

**Given :** x and y stand for number of experienced person and fresh workmen respectively.

Total number of units of work done by experienced person per day is 5x.

Total number of units of work done by fresh one per day is 3y.

Total number of units of work done by both experienced person and fresh one per day is

= 5x + 3y

As per the question, total number of units of work per day should be at least 30 units.

That is, total number of units of work (5x + 3y) should be equal to 30 or more than 30.

So, this situation can be expressed as

5x + 3y ≥ 30

**Example 4 :**

The rules and regulations demand that the employer should employ not more than 5 experienced hands (x) to 1 fresh one (y). How can this fact be expressed ?

**Solution :**

**Given :** x and y stand for number of experienced person and fresh workmen respectively.

As per the question, no. of experienced hands(x) should not be more than 5

That is, no. of experienced hands should be equal to 5 or less than 5

So, we have

x ≤ 5 or x/5 ≤ 1 -----(1)

According to the question, no. of fresh hands is equal to 1.

So, we have

y = 1

Substitute y for z in (1).

(1)-----> x/5 ≤ y

Multiply each side by 5.

x ≤ 5y (or) 5y ≥ x

So, the situation can be expressed as

x ≤ 5y (or) 5y ≥ x

**Example 5 :**

The union however forbids the employer to employ less than 2 experienced persons (x) to each fresh person (y). How can this situation be expressed ?

**Solution :**

**Given :** x and y stand for number of experienced person and fresh workmen respectively.

In this problem, the word "forbid" plays an important role.

Meaning of "Forbid" is "Not allowed"

The union forbids the employer to employ less than 2 experienced hands.

That is, the union does not allow the employer to employ less than 2 experienced hands.

Therefore, the employer should employ 2 or more than 2 experienced hands.

Then, we have

x ≥ 2 (or) x/2 ≥ 1 -----(1)

And also, no. of fresh persons to be employed is equal to 1

Then, we have

y = 1

Substitute y for 1 in (1).

(1)-----> x / 2 ≥ y (or) y ≤ x / 2

So, the situation can be expressed as

x / 2 ≥ y (or) y ≤ x / 2

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems using linear inequalities.

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