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 lessthan 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 lessthansymbol, illuminating its essence, exploring its applications in mathematical expressions, and elucidating its realworld implications. By the end of this journey, you will not only comprehend the mechanics of the lessthan 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 threefourths.

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.
RealLife 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.

Recipe Adjustment:
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 10yearold 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 lessthan 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 realworld 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 problemsolving.
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