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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.

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