Solving Linear Inequalities and Interval Notation

Linear Inequalities and Interval Notation

Introduction to Linear Inequalities

Linear inequalities are mathematical statements that compare linear expressions using inequality symbols such as >, <, ≥, and ≤. The solution to a linear inequality is a set of values that make the inequality true, often represented on a number line and using interval notation.

Solving and Representing Linear Inequalities

• Standard Notation: The inequality is written using symbols (e.g., x > 2).

• Number Line Representation: Solutions are shown as arrows or shaded regions on a number line, indicating which values of x satisfy the inequality.

• Interval Notation: Solutions are expressed as intervals, using parentheses ( ) for open endpoints and brackets [ ] for closed endpoints.

Examples and Interval Notation

Inequality (Standard Notation)	Number Line Representation	Interval Notation
x > 2	Arrow to the right from 2 (open circle)	(2, infty)
x < 2	Arrow to the left from 2 (open circle)	(-infty, 2)
x ≥ 2	Arrow to the right from 2 (closed circle)	[2, infty)
x ≤ 2	Arrow to the left from 2 (closed circle)	(-infty, 2]
-2 < x < 2	Arrow between -2 and 2 (open circles)	(-2, 2)
-2 ≤ x ≤ 2	Arrow between -2 and 2 (closed circles)	[-2, 2]

Key Concepts and Definitions

• Open Interval: $(a, b)$ means all values between a and b, not including endpoints.

• Closed Interval: $[a, b]$ means all values between a and b, including endpoints.

• Infinity: $( ext{or } -infty)$ is always paired with a parenthesis, never a bracket, because infinity is not a specific number.

• Compound Inequality: An inequality that combines two statements, such as $-2 < x < 2$.

Extra Notes

• Intervals ending in $infty$: Use a parenthesis, e.g., $(2, infty)$.

• Intervals starting with $-infty$: Use a parenthesis, e.g., $(-infty, 2)$.

• Compound inequalities: Represented as intervals, e.g., $-2 < x < 2$ is $(-2, 2)$.

Application Example

• Example: Solve $x geq 3$ and write the solution in interval notation. Solution: The solution is all real numbers greater than or equal to 3. Interval Notation: $[3, infty)$

Additional info: Interval notation is a concise way to describe sets of numbers and is widely used in algebra to express solution sets for inequalities.