A Statement Using Symbols To Compare Numbers Explained

A Statement Using Symbols To Compare Numbers Is Called a mathematical inequality. COMPARE.EDU.VN explains that these inequalities are fundamental in mathematics, providing a way to express relationships between values that are not necessarily equal. Understanding how to interpret and manipulate inequalities is essential for problem-solving across various fields. These comparisons use less than, greater than, less than or equal to, and greater than or equal to.

1. Understanding Mathematical Inequalities: An In-Depth Exploration

Mathematical inequalities are statements that compare two expressions, indicating that they are not necessarily equal. These comparisons are crucial in mathematics and various fields, offering a way to describe relationships between values. Instead of asserting that two values are identical, inequalities specify whether one value is greater than, less than, or within a certain range of another. This section explores the fundamental aspects of mathematical inequalities, including their definition, symbols, and practical applications, while highlighting the resources available at COMPARE.EDU.VN.

1.1 Definition of Mathematical Inequalities

A mathematical inequality is a statement that compares two expressions using inequality symbols. Unlike equations, which assert the equality of two expressions, inequalities describe a range of possible values or relationships between values that are not precisely the same. Inequalities are used to express conditions where one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity. These comparisons are essential in various areas of mathematics, including algebra, calculus, and analysis.

1.2 Common Inequality Symbols

Understanding the symbols used in mathematical inequalities is crucial for interpreting and solving them. The primary symbols used are:

> (Greater Than): Indicates that the value on the left side of the symbol is greater than the value on the right side. For example, 5 > 3 means that 5 is greater than 3.
< (Less Than): Indicates that the value on the left side of the symbol is less than the value on the right side. For example, 2 < 7 means that 2 is less than 7.
≥ (Greater Than or Equal To): Indicates that the value on the left side of the symbol is greater than or equal to the value on the right side. For example, x ≥ 4 means that x can be 4 or any value greater than 4.
≤ (Less Than or Equal To): Indicates that the value on the left side of the symbol is less than or equal to the value on the right side. For example, y ≤ 10 means that y can be 10 or any value less than 10.
≠ (Not Equal To): Indicates that the value on the left side of the symbol is not equal to the value on the right side. For example, a ≠ b means that a is not equal to b.

1.3 Types of Inequalities

Mathematical inequalities can be classified into different types based on their complexity and the types of expressions they involve. The main types include:

Linear Inequalities: These involve linear expressions and can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants, and x is a variable.
Polynomial Inequalities: These involve polynomial expressions and can include quadratic inequalities (e.g., x² + 3x – 4 > 0) and higher-degree inequalities.
Rational Inequalities: These involve rational expressions (fractions with polynomials in the numerator and denominator). Solving rational inequalities requires careful consideration of the values that make the denominator zero, as these values are not included in the solution.
Absolute Value Inequalities: These involve absolute value expressions and can be written in the form |x| < a, |x| > a, |x| ≤ a, or |x| ≥ a, where a is a constant.

1.4 Properties of Inequalities

Understanding the properties of inequalities is essential for manipulating and solving them. Key properties include:

Addition and Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the direction of the inequality. If a < b, then a + c < b + c and a – c < b – c.
Multiplication and Division Property:
- Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality. If a 0, then ac < bc and a/c < b/c.
- Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality. If a bc and a/c > b/c.
Transitive Property: If a < b and b < c, then a < c. This property allows for the comparison of multiple values in a sequence.

1.5 Solving Inequalities

Solving inequalities involves finding the range of values that satisfy the inequality. The process typically involves the following steps:

Simplify the Inequality: Combine like terms and simplify the expressions on both sides of the inequality.
Isolate the Variable: Use the addition, subtraction, multiplication, and division properties to isolate the variable on one side of the inequality. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
Express the Solution: Write the solution in interval notation or graph it on a number line to represent all values that satisfy the inequality.

1.6 Practical Applications of Inequalities

Mathematical inequalities have numerous practical applications in various fields, including:

Economics: Inequalities are used to model constraints and optimization problems. For example, they can represent budget constraints, production limits, and market equilibrium conditions.
Engineering: Inequalities are used in designing systems and structures to ensure safety and reliability. They can represent tolerance levels, stress limits, and performance criteria.
Computer Science: Inequalities are used in algorithm analysis and optimization. They can represent time complexity, space complexity, and resource allocation constraints.
Statistics: Inequalities are used in hypothesis testing and confidence intervals. They can represent significance levels, error bounds, and probability ranges.
Everyday Life: Inequalities are used in making decisions and comparisons in everyday situations. For example, they can help determine if a budget is sufficient for a purchase, if a speed is within the legal limit, or if a weight is within a safe range.

1.7 Inequalities in Real-World Scenarios

Consider the following real-world scenarios where inequalities are applied:

Budgeting: A person wants to ensure that their monthly expenses do not exceed their income. They can use the inequality E ≤ I, where E represents total expenses and I represents total income.
Speed Limits: A driver needs to adhere to the speed limit on a highway. They can use the inequality v ≤ 65, where v represents the vehicle’s speed in miles per hour.
Temperature Range: A scientist needs to maintain a chemical reaction within a specific temperature range. They can use the inequality 20°C ≤ T ≤ 30°C, where T represents the temperature in degrees Celsius.
Fitness Goals: An athlete wants to ensure they are burning enough calories during a workout. They can use the inequality C ≥ 500, where C represents the number of calories burned.

1.8 COMPARE.EDU.VN: Your Resource for Understanding Inequalities

COMPARE.EDU.VN offers a wealth of resources to help you understand and master mathematical inequalities. Our website provides detailed explanations, examples, and practice problems to enhance your understanding. Whether you are a student learning the basics or a professional applying inequalities in your field, COMPARE.EDU.VN is your go-to resource for reliable and comprehensive information.

1.9 Advanced Topics in Inequalities

For those seeking a deeper understanding of mathematical inequalities, advanced topics include:

Calculus: Inequalities are used to define limits, continuity, and differentiability. They are also used in optimization problems to find maximum and minimum values.
Real Analysis: Inequalities play a fundamental role in proving theorems and establishing properties of real numbers and functions.
Functional Analysis: Inequalities are used to define norms, inner products, and distances in vector spaces.
Optimization Theory: Inequalities are used to formulate and solve optimization problems, including linear programming, nonlinear programming, and dynamic programming.

1.10 The Importance of Understanding Inequalities

Understanding mathematical inequalities is crucial for problem-solving and decision-making in various fields. By mastering the concepts and techniques related to inequalities, you can enhance your analytical skills and improve your ability to tackle complex problems. Whether you are analyzing financial data, designing engineering systems, or optimizing computer algorithms, inequalities provide a powerful tool for understanding and managing relationships between values.

2. Symbols Used in Inequality Statements: A Comprehensive Guide

In mathematics, inequality statements use specific symbols to compare numbers and expressions. Understanding these symbols is crucial for interpreting and solving inequalities accurately. This section provides a comprehensive guide to the symbols used in inequality statements, along with examples and explanations to enhance understanding, utilizing the resources available at COMPARE.EDU.VN.

2.1 The Greater Than Symbol (>)

The “greater than” symbol (>) indicates that the value on the left side of the symbol is larger than the value on the right side. It is used to express that one quantity exceeds another.

Example: 7 > 3 (7 is greater than 3)

This statement means that the number 7 is larger than the number 3.
Application: In real-world scenarios, this could represent a situation where one item costs more than another, or one person’s score is higher than another’s.

2.2 The Less Than Symbol (<)

The “less than” symbol (<) indicates that the value on the left side of the symbol is smaller than the value on the right side. It is used to express that one quantity is exceeded by another.

Example: 4 < 9 (4 is less than 9)

This statement means that the number 4 is smaller than the number 9.
Application: This could represent a scenario where one item weighs less than another, or one person’s age is younger than another’s.

2.3 The Greater Than or Equal To Symbol (≥)

The “greater than or equal to” symbol (≥) indicates that the value on the left side of the symbol is either larger than or equal to the value on the right side. It combines the concepts of “greater than” and “equal to.”

Example: x ≥ 5 (x is greater than or equal to 5)

This statement means that x can be 5 or any number larger than 5.
Application: In practical terms, this could represent a minimum requirement, such as a minimum age to participate in an activity.

2.4 The Less Than or Equal To Symbol (≤)

The “less than or equal to” symbol (≤) indicates that the value on the left side of the symbol is either smaller than or equal to the value on the right side. It combines the concepts of “less than” and “equal to.”

Example: y ≤ 10 (y is less than or equal to 10)

This statement means that y can be 10 or any number smaller than 10.
Application: This could represent a maximum limit, such as a maximum weight allowed in an elevator.

2.5 The Not Equal To Symbol (≠)

The “not equal to” symbol (≠) indicates that the value on the left side of the symbol is not the same as the value on the right side. It is used to express that two quantities are different.

Example: a ≠ b (a is not equal to b)

This statement means that a and b have different values.
Application: This could represent a scenario where two items are different prices, or two people have different heights.

2.6 Combining Inequality Symbols

Inequality symbols can be combined to express more complex relationships. For example:

Compound Inequalities: These combine two inequalities into one statement.
- Example: 3 < x < 7 (x is greater than 3 and less than 7)
- This means x is between 3 and 7, not including 3 and 7.
- Example: 1 ≤ y ≤ 5 (y is greater than or equal to 1 and less than or equal to 5)
- This means y is between 1 and 5, including 1 and 5.

2.7 Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute value, which represents the distance of a number from zero.

Example: |x| < 3 (the absolute value of x is less than 3)

This means x is between -3 and 3, or -3 < x < 3.
Example: |x| > 2 (the absolute value of x is greater than 2)

This means x is either less than -2 or greater than 2, or x < -2 or x > 2.

2.8 Practical Applications of Inequality Symbols

Understanding and using inequality symbols is essential in various practical applications:

Economics: Inequalities are used to represent constraints, such as budget limits or production capacities.
Engineering: Inequalities are used to define tolerance levels and safety margins.
Computer Science: Inequalities are used in algorithm analysis to compare the efficiency of different algorithms.
Statistics: Inequalities are used in hypothesis testing to determine the significance of results.
Everyday Life: Inequalities are used in making decisions, such as comparing prices, determining eligibility, and setting limits.

2.9 Examples in Real-World Scenarios

Consider the following real-world scenarios where inequality symbols are applied:

Budgeting: A person wants to spend no more than $100 on groceries. They can use the inequality C ≤ 100, where C represents the cost of groceries.
Speed Limits: A driver must maintain a speed of at least 45 mph but no more than 65 mph on a highway. They can use the compound inequality 45 ≤ v ≤ 65, where v represents the vehicle’s speed.
Temperature Control: A scientist needs to keep a reaction at a temperature between 20°C and 30°C. They can use the compound inequality 20 ≤ T ≤ 30, where T represents the temperature.
Eligibility: To be eligible for a program, a person must be at least 18 years old. This can be represented as A ≥ 18, where A represents the person’s age.

2.10 COMPARE.EDU.VN: Your Resource for Inequality Symbols

COMPARE.EDU.VN provides comprehensive resources to help you master the use of inequality symbols. Our website offers detailed explanations, examples, and practice problems to reinforce your understanding. Whether you are a student learning the basics or a professional applying inequalities in your work, COMPARE.EDU.VN is your trusted resource for reliable information.

3. Types of Mathematical Inequalities: A Detailed Overview

Mathematical inequalities come in various forms, each with its own characteristics and methods for solving. Understanding the different types of inequalities is crucial for effectively tackling mathematical problems. This section provides a detailed overview of the primary types of mathematical inequalities, including linear, polynomial, rational, and absolute value inequalities, with resources from COMPARE.EDU.VN.

3.1 Linear Inequalities

Linear inequalities are inequalities that involve linear expressions. A linear expression is an expression in which the highest power of the variable is 1. Linear inequalities can be written in the forms ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants, and x is a variable.

Example: 2x + 3 > 7

To solve this linear inequality:
1. Subtract 3 from both sides: 2x > 4
2. Divide both sides by 2: x > 2
3. The solution is x > 2, which means x can be any number greater than 2.
Graphical Representation: Linear inequalities can be represented graphically on a number line. For the example x > 2, the number line would show an open circle at 2 and a line extending to the right, indicating all values greater than 2.

3.2 Polynomial Inequalities

Polynomial inequalities involve polynomial expressions. A polynomial expression is an expression that consists of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Common types of polynomial inequalities include quadratic inequalities (involving quadratic expressions) and higher-degree inequalities.

Quadratic Inequality Example: x² – 3x – 4 > 0

To solve this quadratic inequality:
1. Factor the quadratic expression: (x – 4)(x + 1) > 0
2. Find the critical points (values of x that make the expression equal to 0): x = 4 and x = -1
3. Test intervals on a number line:
 - For x < -1, (x – 4) is negative and (x + 1) is negative, so (x – 4)(x + 1) is positive.
 - For -1 < x < 4, (x – 4) is negative and (x + 1) is positive, so (x – 4)(x + 1) is negative.
 - For x > 4, (x – 4) is positive and (x + 1) is positive, so (x – 4)(x + 1) is positive.
4. The solution is x < -1 or x > 4.
Higher-Degree Inequality Example: x³ – 6x² + 11x – 6 < 0

To solve this higher-degree inequality:
1. Factor the polynomial: (x – 1)(x – 2)(x – 3) < 0
2. Find the critical points: x = 1, x = 2, and x = 3
3. Test intervals on a number line to determine where the expression is negative.
4. The solution is x < 1 or 2 < x < 3.

3.3 Rational Inequalities

Rational inequalities involve rational expressions, which are fractions with polynomials in the numerator and denominator. Solving rational inequalities requires careful consideration of the values that make the denominator zero, as these values are not included in the solution.

Example: (x + 2) / (x – 3) > 0

To solve this rational inequality:
1. Find the critical points (values of x that make the numerator or denominator equal to 0): x = -2 and x = 3
2. Test intervals on a number line:
 - For x < -2, (x + 2) is negative and (x – 3) is negative, so (x + 2) / (x – 3) is positive.
 - For -2 < x < 3, (x + 2) is positive and (x – 3) is negative, so (x + 2) / (x – 3) is negative.
 - For x > 3, (x + 2) is positive and (x – 3) is positive, so (x + 2) / (x – 3) is positive.
3. The solution is x < -2 or x > 3. Note that x cannot be equal to 3 because the denominator would be zero.

3.4 Absolute Value Inequalities

Absolute value inequalities involve absolute value expressions. The absolute value of a number is its distance from zero. Absolute value inequalities can be written in the forms |x| < a, |x| > a, |x| ≤ a, or |x| ≥ a, where a is a constant.

Example: |x| < 4

To solve this absolute value inequality:
1. Rewrite the inequality as a compound inequality: -4 < x < 4
2. The solution is -4 < x < 4, which means x is between -4 and 4.
Example: |x| > 2

To solve this absolute value inequality:
1. Rewrite the inequality as two separate inequalities: x < -2 or x > 2
2. The solution is x < -2 or x > 2, which means x is either less than -2 or greater than 2.
Example: |2x – 1| ≤ 5

To solve this absolute value inequality:
1. Rewrite the inequality as a compound inequality: -5 ≤ 2x – 1 ≤ 5
2. Add 1 to all parts of the inequality: -4 ≤ 2x ≤ 6
3. Divide all parts by 2: -2 ≤ x ≤ 3
4. The solution is -2 ≤ x ≤ 3, which means x is between -2 and 3, including -2 and 3.

3.5 Practical Applications of Different Types of Inequalities

Different types of inequalities are used in various practical applications:

Linear Inequalities: Used in budgeting and resource allocation problems.
Polynomial Inequalities: Used in optimization problems, such as finding the maximum profit or minimum cost.
Rational Inequalities: Used in rate and ratio problems, such as determining the maximum speed or minimum time.
Absolute Value Inequalities: Used in error analysis and tolerance testing.

3.6 Examples in Real-World Scenarios

Consider the following real-world scenarios where different types of inequalities are applied:

Budgeting (Linear Inequality): A person wants to ensure their monthly expenses (E) do not exceed their income (I), where E = 500 + 0.3I. The inequality is 500 + 0.3I ≤ I.
Profit Maximization (Quadratic Inequality): A company wants to maximize its profit (P) based on the number of units sold (x), where P = -x² + 10x – 9. The inequality is -x² + 10x – 9 > 0.
Chemical Reaction (Rational Inequality): A scientist needs to maintain the concentration of a chemical (C) above a certain level, where C = (t + 1) / (t – 2) and C > 1.
Manufacturing Tolerance (Absolute Value Inequality): A manufacturer needs to ensure that the diameter of a part (d) is within a tolerance of 0.01 inches of the specified diameter (D), so |d – D| ≤ 0.01.

3.7 COMPARE.EDU.VN: Your Resource for Understanding Inequality Types

COMPARE.EDU.VN offers extensive resources to help you understand and master the different types of mathematical inequalities. Our website provides detailed explanations, examples, and practice problems to reinforce your understanding. Whether you are a student learning the basics or a professional applying inequalities in your field, COMPARE.EDU.VN is your go-to resource for reliable and comprehensive information.

4. Properties of Inequalities: Essential Rules and Guidelines

Understanding the properties of inequalities is essential for manipulating and solving them correctly. These properties allow you to perform operations on inequalities while maintaining the validity of the relationship. This section provides a comprehensive guide to the essential properties of inequalities, including addition, subtraction, multiplication, division, and transitive properties, with resources from COMPARE.EDU.VN.

4.1 Addition Property of Inequalities

The addition property of inequalities states that adding the same number to both sides of an inequality does not change the direction of the inequality.

Rule: If a < b, then a + c < b + c.
Example: If x – 3 < 5, then x – 3 + 3 < 5 + 3, which simplifies to x < 8.
Explanation: Adding the same number to both sides maintains the balance of the inequality, preserving the relationship between the two sides.

4.2 Subtraction Property of Inequalities

The subtraction property of inequalities states that subtracting the same number from both sides of an inequality does not change the direction of the inequality.

Rule: If a < b, then a – c < b – c.
Example: If x + 2 > 7, then x + 2 – 2 > 7 – 2, which simplifies to x > 5.
Explanation: Subtracting the same number from both sides maintains the balance of the inequality, preserving the relationship between the two sides.

4.3 Multiplication Property of Inequalities

The multiplication property of inequalities has two parts, depending on whether you are multiplying by a positive or a negative number.

Multiplying by a Positive Number: If a 0, then ac < bc.
- Example: If x / 3 < 4, then (x / 3) 3 < 4 3, which simplifies to x < 12.
- Explanation: Multiplying both sides by a positive number maintains the direction of the inequality.
Multiplying by a Negative Number: If a bc.
- Example: If -2x < 6, then (-2x) / -2 > 6 / -2, which simplifies to x > -3.
- Explanation: Multiplying (or dividing) both sides by a negative number reverses the direction of the inequality.

4.4 Division Property of Inequalities

The division property of inequalities also has two parts, depending on whether you are dividing by a positive or a negative number.

Dividing by a Positive Number: If a 0, then a / c < b / c.
- Example: If 4x < 16, then (4x) / 4 < 16 / 4, which simplifies to x < 4.
- Explanation: Dividing both sides by a positive number maintains the direction of the inequality.
Dividing by a Negative Number: If a b / c.
- Example: If -3x > 9, then (-3x) / -3 < 9 / -3, which simplifies to x < -3.
- Explanation: Dividing both sides by a negative number reverses the direction of the inequality.

4.5 Transitive Property of Inequalities

The transitive property of inequalities states that if one value is less than another, and that second value is less than a third, then the first value is also less than the third.

Rule: If a < b and b < c, then a < c.
Example: If x < y and y < 5, then x < 5.
Explanation: This property allows you to compare multiple values in a sequence, establishing a relationship between the first and last values.

4.6 Practical Applications of Inequality Properties

Understanding and applying the properties of inequalities is essential in various practical applications:

Economics: Used to analyze budget constraints and resource allocation.
Engineering: Used to define tolerance levels and safety margins in designs.
Computer Science: Used in algorithm analysis to compare the efficiency of different algorithms.
Statistics: Used in hypothesis testing to determine the significance of results.
Everyday Life: Used in making decisions, such as comparing prices and determining eligibility.

4.7 Examples in Real-World Scenarios

Consider the following real-world scenarios where the properties of inequalities are applied:

Budgeting: A person wants to ensure their expenses (E) plus savings (S) do not exceed their income (I). Using the addition property, if E + S < I, then E < I – S.
Temperature Control: A scientist needs to maintain a chemical reaction at a temperature (T) that is higher than a minimum (M) and lower than a maximum (X). Using the transitive property, if M < T and T < X, then M < X.
Manufacturing Tolerance: A manufacturer needs to ensure that the length of a part (L) is within a tolerance of 0.01 inches of the specified length (S). If L > S – 0.01 and L < S + 0.01, then S – 0.01 < S + 0.01.
Fitness Goals: An athlete wants to burn more calories (C) than they consume (F) to lose weight. If C > F, and they increase their calorie burn by B, then C + B > F + B (using the addition property).

4.8 COMPARE.EDU.VN: Your Resource for Inequality Properties

COMPARE.EDU.VN offers comprehensive resources to help you master the properties of mathematical inequalities. Our website provides detailed explanations, examples, and practice problems to reinforce your understanding. Whether you are a student learning the basics or a professional applying inequalities in your work, COMPARE.EDU.VN is your trusted resource for reliable information.

4.9 Summary of Inequality Properties

Here is a summary of the properties of inequalities:

Property	Rule	Example
Addition	If a < b, then a + c < b + c	If x – 2 < 5, then x < 7
Subtraction	If a < b, then a – c < b – c	If x + 3 > 8, then x > 5
Multiplication (Positive)	If a < b and c > 0, then ac < bc	If x / 4 < 2, then x < 8
Multiplication (Negative)	If a < b and c < 0, then ac > bc	If -2x < 10, then x > -5
Division (Positive)	If a < b and c > 0, then a / c < b / c	If 5x < 25, then x < 5
Division (Negative)	If a < b and c < 0, then a / c > b / c	If -3x > 12, then x < -4
Transitive	If a < b and b < c, then a < c	If x < y and y < 10, then x < 10

5. Solving Inequalities: Step-by-Step Guide with Examples

Solving inequalities involves finding the range of values that satisfy the inequality. This process requires applying the properties of inequalities to isolate the variable and determine the solution set. This section provides a step-by-step guide on how to solve various types of inequalities, along with detailed examples and explanations, utilizing the resources available at compare.edu.vn.

5.1 Solving Linear Inequalities

Linear inequalities involve linear expressions, and solving them is similar to solving linear equations, with the key difference being the inequality sign.

Step 1: Simplify the Inequality

Combine like terms and simplify both sides of the inequality.
Step 2: Isolate the Variable

Use addition and subtraction to move constants to one side and the variable terms to the other side.
Step 3: Solve for the Variable

Use multiplication or division to isolate the variable. Remember to reverse the inequality sign if multiplying or dividing by a negative number.
Step 4: Express the Solution

Write the solution in interval notation or graph it on a number line.
Example: Solve 3x + 2 < 8
1. Subtract 2 from both sides: 3x < 6
2. Divide both sides by 3: x < 2
3. The solution is x < 2, which can be written in interval notation as (-∞, 2).

5.2 Solving Polynomial Inequalities

Polynomial inequalities involve polynomial expressions, and solving them requires finding the critical points and testing intervals.

Step 1: Rewrite the Inequality

If necessary, rewrite the inequality so that one side is zero.
Step 2: Factor the Polynomial

Factor the polynomial expression.
Step 3: Find the Critical Points

Determine the values of the variable that make the polynomial equal to zero. These are the critical points.
Step 4: Test Intervals

Create a number line and mark the critical points. Choose test values from each interval and plug them into the original inequality to determine if the interval satisfies the inequality.
Step 5: Express the Solution

Write the solution in interval notation, including the intervals that satisfy the inequality.
Example: Solve x² – 5x + 6 > 0
1. Factor the quadratic expression: (x – 2)(x – 3) > 0
2. Find the critical points: x = 2 and x = 3
3. Test intervals on a number line:
 - For x < 2, (x – 2) is negative and (x – 3) is negative, so (x – 2)(x – 3) is positive.
 - For 2 < x < 3, (x – 2) is positive and (x – 3) is negative, so (x – 2)(x – 3) is negative.
 - For x > 3, (x – 2) is positive and (x – 3) is positive, so (x – 2)(x – 3) is positive.
4. The solution is x < 2 or x > 3, which can be written in interval notation as (-∞, 2) ∪ (3, ∞).

5.3 Solving Rational Inequalities

Rational inequalities involve rational expressions, and solving them requires finding the critical points and considering the values that make the denominator zero.

Step 1: Rewrite the Inequality

If necessary, rewrite the inequality so that one side is zero.
Step 2: Find the Critical Points

Determine the values of the variable that make the numerator or denominator equal to zero.
Step 3: Test Intervals

Create a number line and mark the critical points. Choose test values from each interval and plug them into the original inequality to determine if the interval satisfies the inequality.
Step 4: Consider the Denominator

Exclude any values that make the denominator equal to zero, as these values are not part of the solution.
Step 5: Express the Solution

Write the solution in interval notation, including the intervals that satisfy the inequality and excluding any values that make the denominator zero.
Example: Solve (x + 1) / (x – 2) > 0
1. Find the critical points: x = -1 (numerator) and x = 2 (denominator)
2. Test intervals on a number line:
 - For x < -1, (x + 1) is negative and (x – 2) is negative, so (x + 1) / (x – 2) is positive.
 - For -1 < x < 2, (x + 1) is positive and (x – 2) is negative, so (x + 1) / (x – 2) is negative.
 - For x > 2, (x + 1) is positive and (x – 2) is positive, so (x + 1) / (x – 2) is positive.
3. Exclude x = 2, as it makes the denominator zero.
4. The solution is x < -1 or x > 2, which can be written in interval notation as (-∞, -1) ∪ (2, ∞).

5.4 Solving Absolute Value Inequalities

Absolute value inequalities involve absolute value expressions, and solving them requires rewriting the inequality as a compound inequality or two separate inequalities.

Case 1: |x| < a (Less Than)

Rewrite the inequality as -a