Chapter 2, Section 5: Solving Linear Inequalities and Compound Inequalities

Welcome to Chapter 2, Section 5! In this section, we'll be building upon our knowledge of linear equations to tackle linear inequalities and compound inequalities. Understanding these concepts is crucial for many areas of mathematics, so let's dive in!

Key Concepts

  • Linear Inequalities: Similar to linear equations, but instead of an equals sign (=), we have inequality symbols such as < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). For example: $2x + 3 < 7$.
  • Solving Linear Inequalities: We solve inequalities much like we solve equations, with one important exception: multiplying or dividing both sides by a negative number reverses the inequality sign.
  • Graphing Inequalities: We represent the solution set of an inequality on a number line. A closed circle (●) indicates that the endpoint is included in the solution (for $\leq$ and $\geq$), while an open circle (○) indicates that the endpoint is not included (for < and >).
  • Interval Notation: A concise way to represent the solution set of an inequality. For example: $x > 3$ is written as $(3, \infty)$, and $x \leq -2$ is written as $(-\infty, -2]$. Remember, we always use parentheses with infinity and negative infinity.
  • Compound Inequalities: Two inequalities joined by "and" or "or".
    • "And" Inequalities: Represent the intersection of the solutions of the two inequalities. For example: $2 < x < 5$, meaning $x$ is greater than 2 AND less than 5. In interval notation: $(2, 5)$.
    • "Or" Inequalities: Represent the union of the solutions of the two inequalities. For example: $x < 1$ or $x > 4$. In interval notation: $(-\infty, 1) \cup (4, \infty)$.

Example 1: Solving a Linear Inequality

Let's solve the inequality: $$3x - 5 \geq 4$$

  1. Add 5 to both sides: $$3x \geq 9$$
  2. Divide both sides by 3: $$x \geq 3$$

Solution in interval notation: $[3, \infty)$. This means that any number greater than or equal to 3 is a solution.

Example 2: Solving a Compound "And" Inequality

Let's solve: $$-3 \leq 2x + 1 < 5$$

  1. Subtract 1 from all parts: $$-4 \leq 2x < 4$$
  2. Divide all parts by 2: $$-2 \leq x < 2$$

Solution in interval notation: $[-2, 2)$. This means that $x$ is greater than or equal to -2 AND less than 2.

Example 3: Solving a Compound "Or" Inequality

Let's solve: $$x + 2 < 0 \text{ or } 3x > 9$$

  1. Solve the first inequality: $$x < -2$$
  2. Solve the second inequality: $$x > 3$$

Solution in interval notation: $(-\infty, -2) \cup (3, \infty)$. This means that $x$ is less than -2 OR greater than 3.

Remember to always check your solutions, especially when dealing with compound inequalities! Practice makes perfect, so work through plenty of examples. Good luck with tonight's class and don't hesitate to ask questions!