2-3 Inequalities Explained

Key Concepts of Inequalities

Inequalities are mathematical statements that show the relationship between two expressions that are not equal. They use symbols like \( < \), \( > \), \( \leq \), and \( \geq \). Understanding inequalities involves several key concepts:

Inequality Symbols: \( < \), \( > \), \( \leq \), and \( \geq \).
Solving Linear Inequalities: Finding the values that satisfy the inequality.
Graphing Inequalities: Representing the solution set on a number line.
Compound Inequalities: Combining two or more inequalities.

Explanation of Each Concept

Understanding these concepts is crucial for solving and interpreting inequalities.

1. Inequality Symbols

The inequality symbols are used to compare two expressions:

\( < \): Less than
\( > \): Greater than
\( \leq \): Less than or equal to
\( \geq \): Greater than or equal to

Example:

\[ 3x + 2 < 11 \]

This inequality states that \( 3x + 2 \) is less than 11.

2. Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality. This is similar to solving equations, but with an important difference: when multiplying or dividing by a negative number, the inequality sign reverses.

Example:

Solve \( 3x + 2 < 11 \):

Subtract 2 from both sides: \( 3x < 9 \).

Divide both sides by 3: \( x < 3 \).

3. Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. Use an open circle for \( < \) and \( > \), and a closed circle for \( \leq \) and \( \geq \).

Example:

Graph \( x < 3 \):

Place an open circle at 3 and shade to the left.

4. Compound Inequalities

Compound inequalities involve two or more inequalities combined with "and" or "or". "And" means the solution must satisfy both inequalities, while "or" means the solution can satisfy either inequality.

Example:

Solve \( 2 < x + 3 \leq 7 \):

Subtract 3 from all parts: \( -1 < x \leq 4 \).

The solution is \( x \) such that \( -1 < x \leq 4 \).

Examples and Analogies

To better understand inequalities, consider the following analogy:

Imagine you are planning a party and you need to invite guests who are older than 12 but younger than 18. This can be represented as the compound inequality \( 12 < x < 18 \). The solution set includes all ages between 12 and 18, not including 12 and 18.

Practical Applications

Inequalities are used in various real-world applications, such as setting budget constraints, determining eligibility criteria, and analyzing data ranges. Understanding inequalities allows for more precise problem-solving and decision-making in these contexts.