Chapter 2 Review: Equations & Inequalities – Study Notes

Equations & Inequalities

Introduction

This chapter focuses on the foundational concepts of equations and inequalities, which are essential for solving mathematical problems in algebra. Understanding how to manipulate and solve these expressions is crucial for success in college algebra and beyond.

Linear Equations

• Definition: A linear equation is an equation of the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable.

• Solution: To solve for $x$, isolate the variable:

  • $ax + b = 0$

  • $ax = -b$

  • $x = \frac{-b}{a}$

• Example: Solve $3x - 7 = 2$.

  • $3x - 7 = 2$

  • $3x = 9$

  • $x = 3$

Quadratic Equations

• Definition: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

• Solution Methods:

  • Factoring

  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  • Completing the Square

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring.

  • $(x - 2)(x - 3) = 0$

  • $x = 2$ or $x = 3$

Other Types of Equations

• Absolute Value Equations: Equations involving $|x|$ require considering both positive and negative cases.

  • Example: $|x - 3| = 5$ leads to $x - 3 = 5$ or $x - 3 = -5$

  • Solutions: $x = 8$ or $x = -2$

• Rational Equations: Equations with variables in the denominator. Multiply both sides by the least common denominator (LCD) to clear fractions.

• Radical Equations: Equations with variables under a root. Isolate the radical and raise both sides to the appropriate power.

Inequalities

• Linear Inequalities: Similar to linear equations, but with inequality signs ($<, >, \leq, \geq$). Remember to reverse the inequality when multiplying or dividing by a negative number.

• Quadratic Inequalities: Solve $ax^2 + bx + c > 0$ (or $<, \leq, \geq$) by finding the zeros and testing intervals.

• Compound Inequalities: Involve two inequalities joined by 'and' or 'or'. Solve each part and find the intersection or union of solutions.

• Absolute Value Inequalities: $|x| < a$ means $-a < x < a$; $|x| > a$ means $x < -a$ or $x > a$.

• Example: Solve $2x - 3 > 7$.

  • $2x > 10$

  • $x > 5$

Applications of Equations and Inequalities

• Word Problems: Translate real-world situations into equations or inequalities, define variables, and solve for the unknowns.

• Example: If a number increased by 7 is 15, what is the number?

  • Let $x$ be the number: $x + 7 = 15$

  • $x = 8$

Summary Table: Types of Equations and Solution Methods

Type of Equation	General Form	Solution Method	Example
Linear	$ax + b = 0$	Isolate $x$	$2x + 3 = 7$
Quadratic	$ax^2 + bx + c = 0$	Factoring, Quadratic Formula, Completing the Square	$x^2 - 4 = 0$
Absolute Value	$|x| = a$	Set $x = a$ and $x = -a$	$|x| = 5$
Rational	$\frac{P(x)}{Q(x)} = 0$	Multiply by LCD, solve numerator	$\frac{x+1}{x-2} = 0$
Radical	$\sqrt{x} = a$	Square both sides	$\sqrt{x} = 3$

Key Properties and Tips

• Check for extraneous solutions, especially when dealing with rational and radical equations.

• Always verify solutions in the original equation or inequality.

• Graphical methods can help visualize solutions, especially for inequalities.

Additional info: The above summary is based on standard college algebra content for equations and inequalities, as the original file referenced review and test problems from Chapter 2 but did not provide explicit content. The structure and examples are inferred from typical textbook coverage of this topic.