Word Problem and Solving Inequalities – JSS2 Mathematics Lesson Note (First Term)

An inequality compares two quantities using the signs <, >, ≤ or ≥. To solve it, you find the values of X that make it true. You add, subtract, multiply or divide both sides by the same quantity, as you do with an equation. Reverse the inequality sign when you multiply or divide by a negative number.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the rules for solving inequalities;
• solve inequalities and find the greatest or smallest integer value of X;
• write word problems as inequalities in X;
• solve word problems that lead to inequalities.

What are the rules for solving inequalities?

1. An inequality remains true when the same quantity is added to or subtracted from both sides.
2. An inequality remains true when both sides are multiplied or divided by the same positive quantity.
3. An inequality remains true when both sides are multiplied or divided by a negative quantity, provided the inequality sign is reversed.

How do you solve an inequality?

Example 1

Find the greatest possible value of X that satisfies the inequality 8 + 2X > 3 + 5X, if X is an integer.

Solution:

8 + 2X > 3 + 5X

Subtract 8 from both sides:

2X > 3 – 8 + 5X

2X > -5 + 5X

Subtract 5X from both sides:

2X – 5X > -5, so -3X > -5

Divide both sides by -3 and reverse the inequality sign (rule 3):

X < 5/3

The greatest integer value of X is 1.

Example 2

Find the smallest integer value of X that satisfies the inequality 7X – 2 ≥ 5X – 6.

Solution:

7X – 2 ≥ 5X – 6

Add 2 to both sides:

7X ≥ 5X – 6 + 2

7X ≥ 5X – 4

Subtract 5X from both sides:

7X – 5X ≥ -4, so 2X ≥ -4

Divide both sides by 2:

X ≥ -2

The smallest integer value of X is -2.

How do word problems lead to inequalities?

Example 1

David is X years old. In 4 years, his age will still be less than 12 years. (a) Write this information as an inequality in X. (b) Find the maximum age of David to the nearest whole number.

Solution:

In 4 years, David’s age will be (X + 4) years.

If at that time his age is less than 12, then:

X + 4 < 12

Subtract 4 from both sides:

X < 12 – 4

X < 8

The maximum age of David is 7 years.

Example 2

A man had ₦X. Out of this, he used ₦1000 to pay his house rent. The amount he had left was not more than ₦3500. (a) Write this information as an inequality in X. (b) Solve for X.

Solution:

The man used ₦1000 to pay his house rent out of ₦X, so the amount left is ₦(X – 1000).

This amount is not more than ₦3500, so:

X – 1000 ≤ 3500

Add 1000 to both sides:

X ≤ 3500 + 1000, so X ≤ 4500

Hence the man had ₦4500 or less.

Evaluation

1. State the three rules for solving inequalities.
2. Solve the inequality 5X – 3 ≥ 2X + 6.
3. Find the greatest integer value of X that satisfies 4 + X > 6 + 3X.
4. Sarah is X years old. In 5 years, her age will still be less than 15 years. Write this as an inequality in X and find her maximum age.

Assignment

1. Find the smallest integer value of X that satisfies 6X – 4 ≥ 2X + 8.
2. A woman had ₦X. She spent ₦2000 on food. The amount she had left was not more than ₦5000. Write this as an inequality in X and solve for X.