Chapter 2, Sections 7 & 8: Mastering Inequalities and Absolute Value

Welcome to Professor Baker's Math Class! This post summarizes the key topics covered in tonight's lesson from Chapter 2, sections 7 and 8. We'll be diving into inequalities and absolute value, so let's get started!

Section 2.7: Solving Inequalities

In this section, we explore how to solve various types of inequalities. Remember that solving inequalities is very similar to solving equations, with one crucial difference: multiplying or dividing by a negative number reverses the inequality sign.

Here are some key concepts:

  • Basic Inequalities: Understanding the symbols < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to).
  • Solving Linear Inequalities: Use the same algebraic techniques as solving equations (addition, subtraction, multiplication, division) while paying attention to the sign reversal rule.
  • Graphing Inequalities: Representing the solution set on a number line. Use open circles for strict inequalities (< or >) and closed circles for inclusive inequalities ($\leq$ or $\geq$).
  • Interval Notation: Expressing the solution set using intervals, such as $(a, b)$, $[a, b]$, $(a, \infty)$, etc. Remember to use parentheses for open intervals and brackets for closed intervals.

Example: Solve and graph the inequality $2x + 3 < 7$.

Solution:

Subtract 3 from both sides: $2x < 4$

Divide both sides by 2: $x < 2$

The solution in interval notation is $(-\infty, 2)$.

Section 2.8: Absolute Value Equations and Inequalities

Absolute value represents the distance of a number from zero. This means that absolute value equations and inequalities often have two possible solutions.

Key concepts to remember:

  • Definition of Absolute Value: $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.
  • Solving Absolute Value Equations: To solve $|ax + b| = c$, set up two equations: $ax + b = c$ and $ax + b = -c$. Solve each equation separately.
  • Solving Absolute Value Inequalities:
    • For $|ax + b| < c$, solve the compound inequality $-c < ax + b < c$.
    • For $|ax + b| > c$, solve the compound inequality $ax + b < -c$ or $ax + b > c$.

Example: Solve the absolute value equation $|2x - 1| = 5$.

Solution:

Set up two equations:

$2x - 1 = 5$ and $2x - 1 = -5$

Solving the first equation:

$2x = 6$, so $x = 3$

Solving the second equation:

$2x = -4$, so $x = -2$

The solutions are $x = 3$ and $x = -2$.

Remember to practice these concepts by working through various problems. Understanding inequalities and absolute value is crucial for success in algebra and beyond. Keep practicing, and you'll master these skills in no time!