Inequalities
Understanding Inequalities
Inequalities are mathematical expressions that describe the relative size or order of two objects. They are used to show that one quantity is larger, smaller, or equal to another quantity within a certain range. Inequalities are fundamental in various fields of mathematics, including algebra, calculus, and optimization.
Basic Inequalities
There are several types of basic inequalities:
- Greater than: $a > b$ means that $a$ is greater than $b$.
- Less than: $a < b$ means that $a$ is less than $b$.
- Greater than or equal to: $a \geq b$ means that $a$ is greater than or equal to $b$.
- Less than or equal to: $a \leq b$ means that $a$ is less than or equal to $b$.
Properties of Inequalities
Inequalities have several important properties that are used to solve and manipulate them:
- Transitive Property: If $a > b$ and $b > c$, then $a > c$.
- Addition/Subtraction: If $a > b$, then $a + c > b + c$ for any $c$.
- Multiplication/Division by a Positive Number: If $a > b$ and $c > 0$, then $ac > bc$.
- Multiplication/Division by a Negative Number: If $a > b$ and $c < 0$, then $ac < bc$.
- Reversal Property: If $a > b$, then $-a < -b$.
Solving Inequalities
To solve inequalities, we often isolate the variable on one side of the inequality sign, just like we do with equations. However, we must remember to reverse the inequality sign when multiplying or dividing by a negative number.
Types of Inequalities
There are more complex inequalities that involve absolute values, quadratic expressions, and rational expressions. Here are some examples:
- Absolute Value Inequalities: $|a| > b$ implies $a > b$ or $a < -b$.
- Quadratic Inequalities: $ax^2 + bx + c > 0$ can be solved by finding the roots of the corresponding equation and analyzing the intervals.
- Rational Inequalities: $\frac{p(x)}{q(x)} > 0$ can be solved by finding the zeros and undefined points and analyzing the intervals.
Table of Differences and Important Points
| Property | Inequality | Equality |
|---|---|---|
| Reflexivity | Not applicable | $a = a$ |
| Symmetry | Not applicable | If $a = b$, then $b = a$ |
| Transitivity | If $a > b$ and $b > c$, then $a > c$ | If $a = b$ and $b = c$, then $a = c$ |
| Addition/Subtraction | If $a > b$, then $a + c > b + c$ | If $a = b$, then $a + c = b + c$ |
| Multiplication/Division by Positive | If $a > b$ and $c > 0$, then $ac > bc$ | If $a = b$ and $c \neq 0$, then $ac = bc$ |
| Multiplication/Division by Negative | If $a > b$ and $c < 0$, then $ac < bc$ | If $a = b$ and $c \neq 0$, then $ac = bc$ |
| Reversal | If $a > b$, then $-a < -b$ | Not applicable |
Examples
Example 1: Basic Inequality
Solve the inequality $3x - 5 > 1$.
[ \begin{align*} 3x - 5 &> 1 \ 3x &> 6 \ x &> 2 \end{align*} ]
Example 2: Absolute Value Inequality
Solve the inequality $|2x + 3| < 5$.
[ \begin{align*} -5 &< 2x + 3 < 5 \ -8 &< 2x < 2 \ -4 &< x < 1 \end{align*} ]
Example 3: Quadratic Inequality
Solve the inequality $x^2 - 4x + 3 < 0$.
First, factor the quadratic expression:
[ (x - 1)(x - 3) < 0 ]
The roots are $x = 1$ and $x = 3$. Test intervals to find where the inequality holds:
- For $x < 1$, the inequality is positive (false).
- For $1 < x < 3$, the inequality is negative (true).
- For $x > 3$, the inequality is positive (false).
So the solution is $1 < x < 3$.
Example 4: Rational Inequality
Solve the inequality $\frac{x - 1}{x + 2} \geq 0$.
Find the zeros and undefined points: $x = 1$ and $x = -2$. Test intervals:
- For $x < -2$, the inequality is negative (false).
- For $-2 < x < 1$, the inequality is positive (true).
- For $x > 1$, the inequality is positive (true).
So the solution is $-2 < x < 1$ or $x > 1$, including $x = 1$.
Inequalities are a vast topic with many nuances and applications. Understanding the basic principles and properties is essential for solving more complex problems and for applications in calculus, optimization, and other areas of mathematics.