Linear Inequalities and Absolute Value Inequalities – Study Notes

Equations and Inequalities

Linear Inequalities and Absolute Value Inequalities

This section explores the methods for solving linear inequalities and absolute value inequalities, as well as the use of interval notation to describe solution sets. Understanding these concepts is essential for analyzing the range of possible solutions to algebraic statements involving inequalities.

Solving an Inequality

• Inequality: A mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥.

• Solution Set: The set of all numbers that make the inequality true.

• To solve an inequality, isolate the variable using algebraic operations, similar to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

• Interval Notation: A concise way to describe sets of numbers, especially solution sets of inequalities.

• Open Interval (a, b): Represents all real numbers between a and b, but not including a or b.

• Closed Interval [a, b]: Represents all real numbers between a and b, including both endpoints.

• Half-Open Intervals: [a, b) or (a, b] include only one endpoint.

Example: The set of numbers greater than 2 and less than or equal to 5 is written as (2, 5].

Parentheses and Brackets in Interval Notation

• Parentheses ( ) indicate that an endpoint is not included in the interval.

• Brackets [ ] indicate that an endpoint is included in the interval.

Finding Intersections and Unions of Two Intervals

• Intersection ( ∩ ): The set of numbers that are in both intervals.

• Union ( ∪ ): The set of numbers that are in either interval or both.

• Steps:

  1. Graph each interval on a number line.

  2. For intersection, identify the overlap between the intervals.

  3. For union, combine all numbers covered by either interval.

Example: The intersection of [1, 3] and (2, 6) is (2, 3]. The union is [1, 6).

Solving Linear Inequalities in One Variable

• Apply the same algebraic techniques as with equations, but reverse the inequality sign when multiplying or dividing both sides by a negative number.

• Express the solution set using interval notation.

Example: Solve $2x - 5 < 7$. Step 1: Add 5 to both sides: $2x < 12$ Step 2: Divide by 2: $x < 6$ Solution: $(-\infty, 6)$

Recognizing Inequalities with No Solution or All Real Numbers as Solutions

• If the variable cancels and the resulting statement is false (e.g., $0 < -3$), there is no solution.

• If the variable cancels and the resulting statement is true (e.g., $0 < 5$), the solution is all real numbers ($(-\infty, \infty)$).

Solving Compound Inequalities

• Compound Inequality: Combines two inequalities, often using "and" (intersection) or "or" (union).

• "And" means both conditions must be true (intersection).

• "Or" means at least one condition must be true (union).

Example: Solve $2 < x + 1 \leq 5$. Step 1: Subtract 1: $1 < x \leq 4$ Solution: $(1, 4]$

Solving Absolute Value Inequalities

• Absolute Value Inequality: An inequality involving the absolute value function $|x|$.

• For $|x| < a$ (where $a > 0$): $-a < x < a$

• For $|x| > a$ (where $a > 0$): $x < -a$ or $x > a$

Example: Solve $|x - 3| \leq 5$. Step 1: $-5 \leq x - 3 \leq 5$ Step 2: Add 3: $-2 \leq x \leq 8$ Solution: $[-2, 8]$