In the intricate tapestry of mathematical expressions, inequalities stand as pillars, providing a means to compare values and unravel the intricate relationships between quantities. Among these symbols, the less-than sign (<) emerges as a fundamental tool, allowing us to articulate and solve inequalities with precision. This guide aims to delve into the nuanced world of the less-than-symbol, illuminating its essence, exploring its applications in mathematical expressions, and elucidating its real-world implications. By the end of this journey, you will not only comprehend the mechanics of the less-than sign but also possess the skills to navigate the vast landscape of inequalities with confidence.

## Understanding the Less Than Sign:

The less than sign symbolized as "<," is a mathematical notation employed to denote an inequality between two values. When we express a<b, we signify that a is numerically inferior to b. This seemingly modest symbol is an indispensable component of mathematical language, enabling us to articulate relationships where one quantity is perceptibly smaller than another.

## Usage in Mathematical Expressions:

• ### Basic Inequalities:

Example: 3<7

Explanation: In this rudimentary example, the less than sign conveys that three is unequivocally less than seven, succinctly represented as 3<7.

• ### Algebraic Inequalities:

Example: 2x<10

Explanation: Transitioning to the realm of algebra, the inequality 2x<10 implies that the variable x must be less than 5 to satisfy the given condition.

• ### Fractional Inequalities:

Example:

Explanation: Introducing fractions, the inequality

underscores that half is indisputably less than three-fourths.

• ### Decimal Inequalities:

Example: 0.3<0.8

Explanation: The application of the less than sign extends seamlessly to decimals, elucidated by 0.3<0.8, highlighting that 0.3 is indeed less than 0.8.

• ### Combining Inequalities:

Example: 2<4 and 4<6

Explanation: By amalgamating inequalities, we can articulate compound statements such as 2<4<6, underscoring the hierarchical nature of numerical relationships.

## Real-Life Word Problems:

• ### Shopping Spree:

Problem: With \$50 at your disposal and each item costing less than \$20, determine the maximum number of items you can purchase.

Solution: Expressing the inequality as 20x<50, where x denotes the number of items, yields x<2.5. Consequently, you can acquire a maximum of two items.

• ### Time Management:

Problem: Given 24 hours in a day and a sleep duration of less than 8 hours, calculate the available time for work and leisure.

Solution: Framing the inequality as 24−x<8, with x representing sleep hours, leads to a surplus of more than 16 hours for work and leisure.

• ### Fuel Efficiency:

Problem: A car covering less than 300 miles on 15 gallons of gas prompts an exploration of its maximum mileage per gallon.

Solution: The inequality

reveals that the car's mileage is less than 20 miles per gallon.

Problem: With a recipe demanding less than 1.5 cups of sugar, ascertain the required quantity for producing 3 batches.

Solution: The inequality 3×x<1.5, where x signifies the cups needed for each batch, establishes that less than 0.5 cups are required per batch.

• ### Temperature Range:

Problem: If the temperature is less than 10 degrees Celsius and decreases by at most 5 degrees, determine the new temperature range.

Solution: The inequality T−5<10 results in T<15, indicating that the new temperature is less than 15 degrees Celsius.

• ### Discount Shopping:

Problem: An item, post a 30% discount, costs less than \$50. Determine the original price. Solution: Expressing the inequality as 0.7×x<50, where x denotes the original price, leads to

• ### Distance and Time:

Problem: If a car travels less than 60 miles per hour, calculate the distance it can cover in 2 hours.

Solution: The inequality

establishes that the car can traverse less than 60 miles in 2 hours.

• ### Growth Rate:

Problem: With a population growing less than 5% annually, predict the maximum population after 3 years, starting from an initial count of 1000.

Solution: The inequality

, with r denoting the growth rate, indicates that the population growth remains less than 5%.

• ### Age Difference:

Problem: A parent is less than twice the age of their 10-year-old child. Determine the maximum age of the parent.

Solution: The inequality 2×

>

, with

=10, leads to the conclusion that the parent's age is less than 20.

• ### Exam Scores:

Problem: A student scores less than 80% on two exams. Calculate the minimum average percentage required to pass.

Solution: Framing the inequality as

<80, with a and b representing exam scores, results in a+b<160. The minimum average needed to pass is less than 80%.

## Conclusion:

In summary, the less-than sign is an indispensable tool in mathematics, offering a nuanced means to express and analyze inequalities. Through our exploration, we have not only deciphered the intricacies of the less than symbol but also applied its principles to a myriad of real-world scenarios. Armed with this understanding, you are now equipped to confidently navigate the intricate landscape of inequalities, bridging the gap between mathematical abstraction and practical problem-solving.

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