Additive Property of Inequality with Whole Numbers

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Introduction

The Additive property of Inequality states that, for any three numbers a, b, and c.

If a > b, then a + c > b + c

If a > b, then a c > b c

Lets start with the simple inequality a > b. If we want to add a quantity c to the left side, we also have to add it to the right side in order to keep the inequality true. We can write this property as

If a > b, then a + c > b + c.

Similarly, if we want to subtract a quantity c from the left side, we also have to subtract it from the right side in order to keep the inequality true. We can write this property as −

If a > b, then a c > b c.

We show one good real-life example to model this property. For instance, suppose that you know two sisters: Angela and Serena. You know that Angela is older than Serena.

So Angelas age > Serenas age.

In say 5 years from now, will Angela still be older than Serena? Of course! Since the sisters are aging the same amount. In algebraic way, you could represent this inequality as −

Angelas age + 5 years > Serenas age + 5 years

Similarly, the inequality comparing the sisters ages 3 years prior to present time would be

Angelas age 3 years > Serenas age 3 years

Example 1

Solve the following using the additive property of inequality −

x 12 > 9

Solution

Step 1:

Given x 12 > 9; using additive property of inequality

We add 12 to both sides

x + 12 12 > 9 + 12; x > 21

Step 2:

So, the solution for the inequality is x > 21

Example 2

Solve the following using the additive property of inequality −

8 x ≥ 13

Solution

Step 1:

Given 8 x ≥ 13; using additive property of inequality

We subtract 8 from both sides

8 x 8 ≥ 13 8; x ≥ 5

Step 2:

Dividing both sides by 1, we get x ≤ 5 after flipping the inequality sign as well.

So, the solution for the inequality is x ≤ 5