How to Balance the Scales: Inequalities with Addition and Subtraction of Mixed Numbers

When dealing with inequalities involving mixed numbers, understanding the impact of addition and subtraction is essential. In this guide, we’ll explore the intricacies of inequalities when adding or subtracting mixed numbers.

Step-by-step Guide:

1. Basics of Inequalities: 

Inequalities use symbols like \(<\) (less than), \(>\) (greater than), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to) to compare two values.

2. Understanding Mixed Numbers: 

A mixed number is a combination of a whole number and a fraction. For instance, \(2 \frac{1}{3}\) is a mixed number.

3. Impact of Addition on Inequalities: 

When you add the same mixed number to both sides of an inequality, the relationship between the two sides remains unchanged.

Example: If \(a > b\), then \(a + c > b + c\) for any mixed number \(c\).

4. Impact of Subtraction on Inequalities: 

Similarly, when you subtract the same mixed number from both sides of an inequality, the relationship remains unchanged.

Example: If \(a < b\), then \(a – c < b – c\) for any mixed number \(c\).

5. Working with Mixed Numbers: 

When adding or subtracting mixed numbers in inequalities:

– First, perform the operation on the whole numbers.

– Then, perform the operation on the fractions. If necessary, find a common denominator for the fractions.

Example 1: 

Given \( 3 \frac{1}{4} < 4 \frac{1}{2} \), what happens when we add \(1 \frac{3}{4}\) to both sides? 

Solution: 

Adding \(1 \frac{3}{4}\) to both sides: 

\[ 3 \frac{1}{4} + 1 \frac{3}{4} < 4 \frac{1}{2} + 1 \frac{3}{4} \] 

\[ 5 < 6 \frac{1}{4} \]

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Example 2: 

Given \( 5 \frac{2}{3} > 4 \frac{1}{3} \), what is the result when we subtract \(2 \frac{1}{3}\) from both sides? 

Solution: 

Subtracting \(2 \frac{1}{3}\) from both sides: 

\[ 5 \frac{2}{3} – 2 \frac{1}{3} > 4 \frac{1}{3} – 2 \frac{1}{3} \] 

\[ 3 \frac{1}{3} > 2 \]

Practice Questions: 

1. If \( 4 \frac{3}{5} > 3 \frac{2}{5} \), what is the inequality after adding \(2 \frac{1}{5}\) to both sides?

2. Given \( 6 \frac{4}{7} < 7 \frac{3}{7} \), what happens when we subtract \(3 \frac{2}{7}\) from both sides?

3. If \( 8 \frac{1}{8} > 7 \frac{3}{8} \), what is the result when we add \(1 \frac{2}{8}\) to both sides?

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Answers: 

1. \( 6 \frac{4}{5} > 5 \frac{3}{5} \)

2. \( 3 \frac{2}{7} < 4 \frac{1}{7} \)

3. \( 9 \frac{3}{8} > 8 \frac{5}{8} \)

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