Applications and Word Problems with Inequalities

The system of inequalities can be defined as the set of at least 2 or more inequalities used when a problem requires multiple solutions and has more than one constraint. The most common way of solving inequality is through graphical representation.

Some real-life applications of the system of inequalities involve determining the number of produced products to maximise the profit, determining the correct combination and composition of drugs to be given to a patient, etc.

Here, we will be focusing on Applications and Word Problems with Inequalities to help students grasp the concept of inequalities and apply that knowledge to solve a given word problem.

Introduction to inequalities and their graphical representation.

• The system of inequalities can be defined as the set of at least 2 or more inequalities used when a problem requires multiple solutions and has more than one constraint. 

• The most common way of solving inequality through representation is by graphical representation.

• The notations of inequalities are read across left to the right, i.e., 5 > 3 is read as five is greater than three and not as three is less than five.

• The meaning of the given symbols in defining a constraint is as follows –

• >, strict inequality, i.e., the left side quantity is greater than the right-side quantity.

• <, strict inequality, i.e., the left side quantity is less than the right-side quantity.

• >, non-strict inequality, the left side quantity is equal to or greater than the right-side quantity. 

• <, non-strict inequality, the left side quantity is equal to or less than the right-side quantity.

• In the graphical representation, we graph a dashed line to represent strict inequalities, whereas we graph a solid line to represent inequalities that are not strict.

Examples of Applications and Word Problems with Inequalities

Let’s focus on the various examples of Applications and Word Problems with Inequalities.

Applications of Inequalities

• Real-life applications of the system of inequalities have a wide range, from determining the number of produced products to maximise the profit to determining the correct combination and composition of drugs to be given to a patient.

• The vertex theorem is generally used to solve real-life problems of inequalities.

• There are also the concepts of linear programming used to solve linear inequalities.

• The system of linear inequalities is useful when a problem requires a wide range of solutions and has more than one constraint.

• Depending on the given constraint, a set of probable solutions can be calculated (or determined).

Word Problems with Inequalities

Solving word problems with inequalities involves a few ordered steps one needs to follow. These steps are –

1. Reading the problem and understanding it.

2. Identify what is given and what we need to find.

3. Express the constraints and information given or to be found in the form of an equation of inequality. (Choose variables and symbols for information and constraint representation.)

4. Solve the inequality.

5. Check the solution regarding the problem and make sure it is conceivable.

6. Write the answer as a whole sentence.

Let us now look at some examples of word problems involving inequalities.

Prob. 1. Emma has received a new job causing her to move. Her monthly salary will be Rs. 35,265. To qualify for renting an apartment, Emma’s monthly salary must be at least three times that of rent value. What is the highest possible rent Emma can pay?

Solution. Let r be the highest rent Emma can pay. We know that Emma’s salary has to be at least three times the rent. Then,

35,265 > 3r

• 11,755 > r

• r < 11,755.

Hence, the highest possible rent Emma can pay is Rs. 11,755.

Prob. 2. Akash and Sameer are in the football team. Last Sunday, Akash scored 3 more goals than Sameer, but they scored less than 9 goals. What is the possible number of goals scored by Akash?

Solution. Let the number of goals scored by Akash and Sameer be x and y.

We know Sunday Akash scored 3 more goals than Sameer. Then,

x = y + 3. – eq. 1.

Also, we know that together they scored less than 9 goals. Then,

x + y < 9. – eq. 2.

Substituting the value of x from eq. 1 in eq. 2, we have,

(y + 3) + y < 9

• 2y < 6

• y < 3.

This means that Sameer has scored less than 3 goals. Thus, Sameer’s possible number of goals are 0, 1, or 2.

Since Akash scored 3 more goals than Sameer, Akash’s possible number of goals are 3, 4, or 5.

Conclusion

By the end of this precise and concise article on Inequalities – Applications and Word Problems with Inequalities –_Maths, CBSE students have come to understand and grasp the concepts of –

• Meaning and representation of a system of inequalities.

• Representation of constraints and their meaning.

• Applications and Word Problems with Inequalities.

• Steps to solve word problems with inequalities.

Examples of Applications and Word Problems with Inequalities have been provided for students to grasp the concepts easily.