Inequalities Comprehensive Notes

Introduction to Inequalities

• Definition: A mathematical expression that shows the relationship between two values that may not be equal. Common symbols include:
  • Greater than: $>$
  • Less than: $<$
  • Greater than or equal to: $$
  • Less than or equal to: $$
  • Not equal to: $
    eq$

Types of Inequalities

• Linear Inequalities

  • Example: $2x - 5 > 3$
  • Solution Technique:
  • Rearrange:
    • $2x > 8$
    • $x > 4$

• Quadratic Inequalities

  • Example: $x^2 - 3x - 10 < 0$
  • Key Steps to Solve:
  • Factor the expression: $ (x - 5)(x + 2) < 0$
  • Identify critical points:
    • $x = 5$ and $x = -2$
  • Number Line Test:
    • Test intervals: $(-, -2)$, $(-2, 5)$, and $(5, )$
    • Determine where the inequality holds true; for example, the solution is $x ext{ between } (-2, 5)$

Rational Inequalities

• Example: \frac{x + 3}{x - 2} \geq 0
• Steps:
  • Identify critical points where the expression is either zero or undefined.
  • Critical points:
  • Denominator $x - 2
    eq 0$ gives $x
    eq 2$
  • Numerator $x + 3 = 0$ gives $x = -3$
  • Number Line Test:
  • Analyze the signs in the intervals: $(-, -3)$, $(-3, 2)$, and $(2, )$

Systems of Inequalities

• Definition: A set of inequalities with common variables.
  • Solve each inequality separately and then find the solution set that satisfies them both.
  • Example:
  • $x > 1$
  • $x ext{ is } ext{ } ext{ and } x ext{ is } ext{ }$

Absolute Value Inequalities

• Concept: Depends on whether it's less than or greater than a particular value.
  • Less than case:
  • If $|ax + b| < c$, then:
    • $-c < ax + b < c$
  • Greater than case:
  • If $|ax + b| > c$, then:
    • $ax + b > c$ or $ax + b < -c$

Graphing Absolute Value Functions

• Basic Structure:
  • Graph of $y = |x|$ looks like a 'V', opening upwards.
  • Transforms when evaluating $y = |ax + b|$ based on linear equations.

Helpful Graphing Tools

• Utilize online resources such as WolframAlpha or Mathway to visualize functions and graphs, aiding in understanding complicated inequalities and their behaviors.