Understanding of inequalities and its properties is important in the field of mathematics. It helps in analysing and optimising systems and processes.

Introduction

Inequalities are fundamental in mathematics, representing relationships where quantities are not equal, introducing concepts of greater than, less than, and their combinations. Understanding them and their properties helps in fields ranging from algebra to real-world applications like finance, engineering, and science.

Inequalities and Properties of Inequalities

This guide will provide an in-depth understanding of this mathematical concept, its types, and key properties.

What is an Inequality?

It is a mathematical expression that shows the relative size or order of two values. It establishes a relationship where one quantity is either greater than, less than, or not equal to another quantity. For example:

$a < b$ , read as a is less than b.
$x \geq y$ , read as x is greater than or equal to y.

Representation

These are represented using symbols like:

$<$ : Less than
$>$ : Greater than
$\leq$ : Less than or equal to
$\geq$ : Greater than or equal to
$\neq$ : Not equal to

Types of Inequalities

Each of their type serve a specific mathematical purpose. A list of them are mentioned here:

Strict Inequality

They use symbols $<\quad or \quad>$ to indicate that one value is strictly less than or greater than another (e.g., $x > y$ ).

Non-Strict Inequality

They use symbols like, $\leq\quad or \quad\geq$ , and allow equality as well (e.g., $x \leq y$ ).

Compound Inequality

It combines two inequalities, and describes a range of values (e.g., $2 < x \leq 5$ ).

Absolute Inequality

This involves absolute values of numbers and shows distance from zero (e.g., $|x| > 5$ implies $x > 5\quad or \quad x < -5$ ).

Properties of Inequalities

Understanding their properties allows us to manipulate and solve them effectively. Here are the main properties:

Addition and Subtraction Properties

For addition:

If $a < b$ , then $a + c < b + c$ for any c.
If $a > b$ , then $a + c > b + c$ for any c.

For subtraction:

If $a < b$ , then $a - c < b - c$ for any c.
If $a > b$ , then $a - c > b - c$ for any c.

Multiplication and Division Properties

For multiplication:

If $a < b\quad and \quad c > 0$ , then $a \cdot c < b \cdot c$
If $a < b\quad and \quad c < 0$ , then $a \cdot c > b \cdot c$ (inequality sign reverses with negative values)

For division:

If $a < b\quad and \quad c > 0$ , then $\quad\quad \frac {a}{c} < \frac {b}{c}$
If $a < b\quad and \quad c < 0$ , then $\quad\quad \frac {a}{c}>\frac {b}{c}$ (inequality sign reverses with negative values)

Transitive Property

If $a < b\quad and\quad b < c,\quad$ then $\quad\quad a < c$ .
If $a \leq b \quad and \quad b \leq c$ , then $a \leq c$ .

Substitution Property

If $a = b$ , then $a$ can replace $b$ in any inequality.

Inverse Property

It involves absolute values and states, if $|x| < a$ , then $-a < x < a$ .

Solving Inequalities

To solve them, follow similar steps as solving equations but pay attention to the properties as well. Here are some examples:

Example 1

Solve $3x + 5 < 20$ .

Solution:

Subtract 5 from both sides:

$3x < 15$

Divide by 3:

$x < 5$

Solution Set:

The solution is x < 5, meaning x can be any number less than 5.

Example 2

Solve $x - 10 < 6x + 20$ .

Solution:

Add 10 to both sides:

$x < 6x + 30$

Subtract 6x from both sides:

$-5x < 30$

$x > -6$

Solution Set:

The solution is $x < -6,$ meaning $x$ can be any number less than –6.

Important Note: When multiplying or dividing by a negative number, reverse the sign.

Applications of Inequalities

These are used in various fields:

Economics: Modeling budgets, profit constraints, and income comparisons.
Engineering: Ensuring safety limits (e.g., stress limits on materials).
Physics: Representing bounds in measurements, such as error margins.
Statistics: Defining confidence intervals and probability ranges.

Conclusion

Inequalities are powerful tools for comparing values and defining ranges. Mastering its properties, helps in solving complex mathematical problems and applying them effectively in real-world contexts.

By understanding how to manipulate them, we can handle everything from simple comparisons to intricate, multi-step problems across diverse applications.

Frequently Asked Questions (FAQs)

What is the difference between a strict and non-strict inequality?

The former (e.g., $<$ ) does not allow equality, while the latter (e.g., $\leq$ ) does.

Why do we reverse the inequality sign when multiplying or dividing by a negative number?

Reversing the sign is mandatory to maintain its truthfulness, since multiplying or dividing by a negative inverts the order of values.

How do absolute inequalities work?

They deal with the distance from zero. For $|x| > a, x$ is either greater than a or less than $-a$ .

Can inequalities have infinite solutions?

Yes, most of them describe ranges of values, often with infinitely many solutions within that range.

How are inequalities represented graphically?

They can be represented on number lines, with shaded areas showing the range of solutions, or on coordinate planes with two variables.

Inequalities and Properties of Inequalities | 101 A Comprehensive Guide

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