## Linear Inequality

February 3, 2021, Basic Mathematics

#### There are a few rules before we begin learning about linear inequalities. Those are:

**Rule1:** Equal numbers can be added to or subtracted from both sides of the inequality.

**Rule 2:** Both sides of an inequality can be multiplied or divided by the same non-zero number.

**Rule 3:** If we divide or multiply any inequality by a negative number, the sign of the inequality will be reversed.

**Now let’s get started with an example and solve it keeping all the above rules in the mind.**

**Ex.1.** George spends $15 from his monthly income for the internet charges. The remaining amount of his monthly income should be at least $250. What would be George’s minimum monthly income?

**Solution:**

Let $x be George’s monthly income.

The remaining amount after spending the internet charges of $15 from the monthly income $x can be written as $(x-15) .

As the remaining amount should be at least $250, this situation can be represented by the inequality as

x-15≥250

To solve this inequality for x, add 15 on both sides of the inequality.

⇒x-15+15≥250+15

⇒x≥265

Thus, George’s minimum monthly income is $265.

**Ex.2. **The flat owner wants to give the apartment on rent to paying guests. He wishes to charge $250 to each person and wants to earn minimum of $1200. How many minimum number of guests have to share the apartment?

**Solution: **

Let x be the number of guests.

The total amount of x guest by collecting $250 from each becomes $ 250 x.

As the total amount is minimum $1200, so it can be written in inequality as

1200<250 x

To solve the inequality for x , dividing both sides by 250

\( \displaystyle \frac{1200}{250} < \frac{250}{250}x \)

⇒4.8< x

The number of guests cannot be decimals, thus consider the next integer that is 5.

Therefore, at least 5 guests have to share the flat to get minimum of total.

**Ex.3.** Jack has $50. He bought a pen of $4 and now he wants to buy apples of cost $3 each. How many maximum apples Jack can buy from the money he has?

**Solution: **

The cost of a pen is $4 and the cost of each apple is $3.

Let xbe the number of maximum apples that Jack can buy.

Total cost of a pen and the cost of x number of apples can be written as $(4+3x).

Jack has $50; this situation can be written by using the inequality as

3x+4 ≤50

To solve this inequality for x , subtracting 4 from both sides

⇒3x+4-4 ≤50-4

⇒3x ≤46

Dividing both sides by 3

⇒ \( \displaystyle \frac{3x}{3} ≤ \frac{46}{3} \)

\( ⇒ \displaystyle x ≤ \frac{46}{3} \)

⇒x ≤15.33

Hence, Jack can but 15 maximum apples.

**Ex. 4:** David has 67 toffees and he wants to distribute some of the toffees to his 7 friends such that each friend has equal number of toffees. What would be the maximum number of toffees that each friend will have?

**Solution:**

Let x be the number of maximum toffees each friend will have.

Thus, 7 friends will have 7x number of toffees.

As David has 67 toffees, so this situation can be written with the inequality as

7x ≤67

To solve this inequality for x , dividing both sides by 7

\( ⇒ \displaystyle \frac{7x}{7} ≤ \frac{67}{7} \)

\( ⇒ \displaystyle x ≤ \frac{67}{7} \)

⇒x≤ 9.57

As the number of toffees should be an integer, each friend will have maximum 9 toffees.