SOLVING ALGEBRAIC AND REAL WORLD PROBLEMS WITH INEQUALITIES

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

So, 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Every month, $35 is withdrawn from Tony’s savings account to pay for his gym membership.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

So, 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

So, 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

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

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