2.7 Linear Inequalities and Absolute Value Inequalities

Linear inequalities are a key concept in algebra, showing how numbers relate to each other. They use symbols like < and > to compare values, and can be represented on number lines or with interval notation.

Absolute value inequalities take this a step further, dealing with the distance from zero. These concepts are crucial for understanding ranges of numbers and solving real-world problems involving limits or boundaries.

Linear Inequalities

Solution sets in interval notation

• Represents a range of real numbers between two endpoints
  • Parentheses () exclude the endpoint from the set
  • Brackets [] include the endpoint in the set
• Uses $-\infty$ for negative infinity and $\infty$ for positive infinity
• $(2, 5]$ all real numbers between 2 and 5, including 5 but not 2
• $[-3, \infty)$ all real numbers greater than or equal to -3

Properties of linear inequalities

• Addition Property: $a < b$, then $a + c < b + c$ for any real number $c$
• Multiplication Property:
  • $a < b$ and $c > 0$, then $ac < bc$
  • $a < b$ and $c < 0$, then $ac > bc$
• Multiplying or dividing an inequality by a negative number reverses the inequality sign (inequality sign reversal)
• Isolate the variable by performing the same operation on both sides, following the properties of inequalities

One-variable inequalities and graphs

• Solve using the properties of inequalities
• Graph the solution set on a number line
  • Open circle (○) endpoint not included in the solution set
  • Closed circle (●) endpoint included in the solution set
  • Shade the portion representing the solution set
    • Shade right if greater than the endpoint
    • Shade left if less than the endpoint

Compound Inequalities and Set Operations

• Compound inequality: a combination of two or more simple inequalities
• Union (∪): the set of all elements that belong to either or both sets
• Intersection (∩): the set of all elements that belong to both sets
• Domain: the set of all possible input values for a function or inequality
• Range: the set of all possible output values for a function or inequality

Absolute Value Inequalities

Absolute value inequalities

• Absolute value is the distance of a number from zero on the number line
  • $|x|$ represents the absolute value of $x$
• Solving $|x| < a$, where $a > 0$:
  1. Solution set is all real numbers between $-a$ and $a$, not including $-a$ and $a$
  2. Interval notation: $(-a, a)$
• Solving $|x| > a$, where $a > 0$:
  1. Solution set is all real numbers less than $-a$ or greater than $a$
  2. Interval notation: $(-\infty, -a) \cup (a, \infty)$
• $|x| \leq a$ or $|x| \geq a$ similar, but endpoints are included
• Solving steps:
  1. Isolate the absolute value expression on one side
  2. Consider two cases: expression inside is positive or negative
  3. Solve each case separately and combine the solution sets