Linear Inequalities

Introduction to Inequalities

Linear inequalities are mathematical statements that compare two expressions using inequality symbols rather than an equals sign. Unlike equations, which have a finite set of solutions, inequalities often have infinitely many solutions represented as intervals on the number line.

• Equation: Uses the equals sign (=) and typically has solutions like .

• Inequality: Uses symbols such as <, >, ≤, ≥ and has solution sets that may include infinitely many values.

Key Inequality Symbols:

• < : Less than

• > : Greater than

• ≤ : Less than or equal to

• ≥ : Greater than or equal to

Differences Between Equations and Inequalities

• Equations: Solutions are specific values (e.g., ).

• Inequalities: Solutions are ranges or intervals (e.g., means all less than 2).

• Solution to an Inequality: Expressed as or , and often written in interval notation.

Interval Notation

Interval notation is a concise way to describe sets of numbers that satisfy an inequality.

• Open Interval: means all such that .

• Closed Interval: means all such that .

• Half-Open Intervals: or

• Infinite Intervals: or

Solving Linear Inequalities

To solve a linear inequality, use similar steps as solving linear equations, but pay special attention to multiplying or dividing by negative numbers, which reverses the inequality sign.

• Step 1: Isolate the variable on one side.

• Step 2: Simplify both sides as needed.

• Step 3: If you multiply or divide both sides by a negative number, reverse the inequality sign.

• Step 4: Express the solution in interval notation and graph it on a number line.

Examples

• Example 1: Solve

  • Add 2 to both sides:

  • Divide by 3:

  • Interval Notation:

• Example 2: Solve

  • Subtract 3:

  • Divide by 2:

  • Interval Notation:

• Example 3: Solve

  • Expand:

  • Subtract :

  • Subtract 7:

  • Rewrite:

  • Interval Notation:

• Example 4: Solve

  • Expand:

  • Combine like terms:

  • Add :

  • Subtract 3:

  • Divide by 9:

  • Interval Notation:

Graphing Solutions

Graphing the solution to an inequality involves marking the interval on a number line. Use an open circle for < or > (not including the endpoint), and a closed circle for ≤ or ≥ (including the endpoint).

Summary Table: Inequality Symbols and Their Meanings

Important Notes

• When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.

• Always check your solution by substituting values from the interval into the original inequality.

Additional info: Some context and steps were inferred to clarify the solving process and interval notation, as the original notes were fragmented.