Which Represents The Solution Set Of The Inequality

Which Represents the Solution Set of the Inequality? A Comprehensive Guide

Understanding how to represent the solution set of an inequality is a fundamental skill in algebra. Inequalities, unlike equations, don't have a single solution but rather a range of solutions. This article will delve deep into various methods of representing these solution sets, exploring different types of inequalities and offering practical examples to solidify your understanding.

Understanding Inequalities

Before we dive into representing solution sets, let's refresh our understanding of inequalities. An inequality is a mathematical statement that compares two expressions using inequality symbols:

• <: Less than
• >: Greater than
• ≤: Less than or equal to
• ≥: Greater than or equal to
• ≠: Not equal to

Inequalities often involve variables, and solving an inequality means finding the values of the variable that make the statement true.

Methods of Representing Solution Sets

There are several ways to represent the solution set of an inequality:

1. Set-Builder Notation

Set-builder notation uses a concise mathematical notation to describe a set. It typically follows this format:

{x | condition}

This reads as "the set of all x such that the condition is true".

Example:

Let's consider the inequality x > 3. In set-builder notation, the solution set would be:

{x | x > 3}

This means the set of all x values such that x is greater than 3.

2. Interval Notation

Interval notation uses parentheses and brackets to represent the range of values in a solution set.

• Parentheses ( ): Indicate that the endpoint is not included in the solution set. Used for strict inequalities (<, >).
• Brackets [ ]: Indicate that the endpoint is included in the solution set. Used for inequalities with "or equal to" (≤, ≥).

Example:

For the inequality x > 3, the interval notation is (3, ∞). The parenthesis at 3 indicates that 3 itself is not part of the solution, while ∞ (infinity) always gets a parenthesis because infinity is not a number.

For the inequality x ≥ 3, the interval notation would be [3, ∞). The bracket at 3 shows that 3 is included.

For the inequality -2 ≤ x < 5, the interval notation is [-2, 5). This represents all numbers between -2 and 5, including -2 but excluding 5.

Common Interval Notation Representations:

• (a, b): Represents all numbers between a and b, excluding a and b.
• [a, b]: Represents all numbers between a and b, including a and b.
• (a, b]: Represents all numbers between a and b, excluding a but including b.
• [a, b): Represents all numbers between a and b, including a but excluding b.
• (a, ∞): Represents all numbers greater than a.
• [a, ∞): Represents all numbers greater than or equal to a.
• (-∞, a): Represents all numbers less than a.
• (-∞, a]: Represents all numbers less than or equal to a.
• (-∞, ∞): Represents all real numbers.

3. Number Line Representation

A number line provides a visual representation of the solution set. We typically plot the critical points on the number line and shade the region representing the solution.

Example:

For the inequality x > 3, we would plot a point at 3, typically using an open circle (o) to indicate that 3 is not included, and then shade the region to the right of 3, representing all values greater than 3.

For the inequality x ≥ 3, we'd use a closed circle (•) at 3 to indicate inclusion and shade to the right.

Solving and Representing Inequalities

Let's work through some examples to illustrate how to solve inequalities and represent their solution sets using different methods.

Example 1: Simple Linear Inequality

Solve the inequality 2x + 5 < 11 and represent the solution set in all three methods.

Solution:

1. Solve the inequality: Subtract 5 from both sides: 2x < 6 Divide both sides by 2: x < 3

2. Set-builder notation: {x | x < 3}

3. Interval notation: (-∞, 3)

4. Number line representation: An open circle at 3 and shading to the left.

Example 2: Inequality with a Negative Coefficient

Solve the inequality -3x + 7 ≥ 13 and represent the solution set.

Solution:

1. Solve the inequality: Subtract 7 from both sides: -3x ≥ 6 Divide both sides by -3. Remember: When dividing or multiplying by a negative number, you must reverse the inequality sign. x ≤ -2

2. Set-builder notation: {x | x ≤ -2}

3. Interval notation: (-∞, -2]

4. Number line representation: A closed circle at -2 and shading to the left.

Example 3: Compound Inequality

Solve the compound inequality -1 ≤ 2x - 3 < 5 and represent the solution set.

Solution:

This is a compound inequality, meaning it involves more than one inequality. We need to solve it in two parts:

1. Solve -1 ≤ 2x - 3: Add 3 to both sides: 2 ≤ 2x Divide by 2: 1 ≤ x or x ≥ 1

2. Solve 2x - 3 < 5: Add 3 to both sides: 2x < 8 Divide by 2: x < 4

Combining the results, we have 1 ≤ x < 4.

2. Set-builder notation: {x | 1 ≤ x < 4}

3. Interval notation: [1, 4)

4. Number line representation: A closed circle at 1, an open circle at 4, and shading between them.

Example 4: Absolute Value Inequality

Solve the absolute value inequality |x - 2| < 3 and represent the solution set.

Solution:

The inequality |x - 2| < 3 means that the distance between x and 2 is less than 3. This can be rewritten as a compound inequality:

-3 < x - 2 < 3

Add 2 to all parts:

-1 < x < 5

2. Set-builder notation: {x | -1 < x < 5}

3. Interval notation: (-1, 5)

4. Number line representation: Open circles at -1 and 5, with shading in between.

Example 5: Quadratic Inequality

Solve the quadratic inequality x² - 4x + 3 > 0 and represent the solution set.

Solution:

First, we solve the quadratic equation x² - 4x + 3 = 0 by factoring: (x - 1)(x - 3) = 0. This gives us roots x = 1 and x = 3.

These roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞). We need to test a value from each interval to determine where the inequality holds true:

• (-∞, 1): Let's test x = 0. 0² - 4(0) + 3 = 3 > 0. This interval satisfies the inequality.
• (1, 3): Let's test x = 2. 2² - 4(2) + 3 = -1 < 0. This interval does not satisfy the inequality.
• (3, ∞): Let's test x = 4. 4² - 4(4) + 3 = 3 > 0. This interval satisfies the inequality.

Therefore, the solution is x < 1 or x > 3.

2. Set-builder notation: {x | x < 1 or x > 3}

3. Interval notation: (-∞, 1) ∪ (3, ∞) (The ∪ symbol means union, combining two separate intervals).

4. Number line representation: Shade the regions to the left of 1 and to the right of 3, using open circles at 1 and 3.

Conclusion

Representing the solution set of an inequality is crucial for fully understanding and communicating the results of your algebraic manipulations. Mastering set-builder notation, interval notation, and number line representations allows you to express solution sets clearly and concisely, providing a comprehensive understanding of the range of values that satisfy the given inequality. Remember to pay close attention to the inequality symbols and the rules for manipulating inequalities, especially when dealing with negative coefficients or absolute values. Practice regularly with different types of inequalities to solidify your skills and build confidence in your ability to solve and represent these important mathematical concepts.