**Solving algebraic and real world problems with inequalities :**

In this section, we will learn how to create an inequality for the given real world problem by using algebraic reasoning and solve for the unknown quantity.

**Example 1 : **

A marine submersible descends more than 40 feet below sea level. As it descends from sea level, the change in elevation is -5 feet per second. For how many seconds does it descend ?

**Solution :**

Let "t" be the number.

**Step 1 : **

From the given information, write an inequality in terms of "t".

Rate of change × Time < Final elevation

-5t < -40

**Step 2 :**

Use Division Property of Inequality.

Divide both sides by -5.

(-5t) / (-5) > (-40) / (-5)

t > 8

Hence, the submersible descends for more than 8 seconds.

**Example 2 : **

Every month, $35 is withdrawn from Tony’s savings account to pay for his gym membership. He has enough savings to withdraw no more than $315. For how many months can Tony pay for his gym membership ?

**Solution :**

Let "m" be the required no. of months.

**Step 1 : **

From the given information, write an inequality in terms of "m".

Rate per month × No. of months

35m ≤ 315

**Step 2 :**

Use Division Property of Inequality.

Divide both sides by 35.

35m / 35 ≤ 315 / 35

m ≤ 9

Hence, Tony can pay for no more than 9 months of his gym membership using this account.

**Example 3 : **

David has scored 110 points in the first level of a game. To play the third level, he needs more than 250 points. To play third level, how many points should he score in the second level ?

**Solution :**

Let "x" be points scored in the second level

**Step 1 : **

He has already had 110 points in the first level.

Points scored scored in the second level = x

Total points in the first two levels = x + 110

**Step 2 :**

Write the inequality.

To play third level, the total points in the first two levels should be more than 250. So, we have

x + 110 > 250

Subtract 110 on from both sides.

(x + 110) - 110 > 250 - 110

x > 140

Hence, he has to score more than 140 points in the second level.

**Example 4 : **

An employer recruits experienced and fresh workmen for his firm under the condition that he cannot employ more then 9 people. If 5 freshmen are recruited, how many experienced men have to be recruited ?

**Solution :**

Let "x" be the no. of freshmen to be recruited.

**Step 1 : **

Write the inequality.

x + 5 ≤ 9

**Step 2 :**

Subtract 5 from both sides.

(x + 5) - 5 ≤ 9 - 5

x ≤ 4

To meet the given condition, no. of freshmen to be recruited can be less than or equal to 4.

**Example 5 : **

An employee of a factory has to maintain an output of at least 30 units of work per week. If there are five working day in a week, how many units of work to be done by him per day ?

**Solution :**

Let "x" be the no. of units of work done per day.

**Step 1 : **

From the given information, we have

Total number of units of work done per week = 5x

**Step 2 :**

Write the inequality.

As per the question, total number of units of work done per week should be at least 30 units. So, we have

5x ≥ 30

Divide both sides by 5

5x/5 ≥ 30/5

x ≥ 6

Hence, the number of units of work to be done per day should be at least 6.

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