Chapter 1 Review: Equations, Inequalities, and Graphs

Equations and Inequalities

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, solving equations is a fundamental skill, especially for linear and quadratic forms.

• Linear Equation: An equation of the form $ax + b = 0$, where $a$ and $b$ are constants.

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$, where $a eq 0$.

• Solution Methods:

  • Isolate the variable using algebraic operations.

  • For quadratics, use factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $2x + 3 = 7$. $2x = 4 \Rightarrow x = 2$

Solving Inequalities

Inequalities compare two expressions using symbols such as <, >, ≤, or ≥. Solutions are often expressed in interval notation or on a number line.

• Linear Inequality: $ax + b < c$

• Quadratic Inequality: $ax^2 + bx + c > 0$

• Solution Methods:

  • Isolate the variable.

  • Test intervals between critical points (roots).

  • Express solutions in interval notation, e.g., $(-\infty, 2)$.

• Example: Solve $x^2 - 4 > 0$. $x^2 - 4 = 0 \Rightarrow x = -2, 2$ Test intervals: $(-\infty, -2)$ and $(2, \infty)$ satisfy the inequality. Solution: $(-\infty, -2) \cup (2, \infty)$

Functions and Graphs

Graphing Polynomial Functions

Polynomial functions are expressions involving powers of $x$. Their graphs can be analyzed for intercepts, turning points, and end behavior.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$

• Intercepts:

  • x-intercepts: Points where $f(x) = 0$.

  • y-intercept: Point where $x = 0$.

• Turning Points: Points where the graph changes direction.

• End Behavior: Determined by the leading term $a_nx^n$.

• Example: The graph shown is a cubic polynomial with three x-intercepts and two turning points.

Graphing Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex: The highest or lowest point, given by $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$.

• Direction: Opens upward if $a > 0$, downward if $a < 0$.

• Example: The graph shown is a parabola opening upward with vertex at $(0,0)$.

Interval Notation and Number Line Representation

Solutions to equations and inequalities are often represented using interval notation and number lines.

• Interval Notation:

  • $(a, b)$: All numbers between $a$ and $b$, not including endpoints.

  • $[a, b]$: All numbers between $a$ and $b$, including endpoints.

  • $(-\infty, a)$: All numbers less than $a$.

  • $(b, \infty)$: All numbers greater than $b$.

• Number Line: Visual representation of solution sets, with open or closed circles indicating inclusion or exclusion of endpoints.

• Example: The number lines shown represent solutions to various inequalities.

Properties of Real Numbers and Radicals

Properties of Real Numbers

Real numbers have several important properties used in algebraic manipulations.

• Commutative Property: $a + b = b + a$

• Associative Property: $(a + b) + c = a + (b + c)$

• Distributive Property: $a(b + c) = ab + ac$

Radical Expressions

Radicals involve roots, such as square roots and cube roots.

• Square Root: $\sqrt{x}$ is a number which, when squared, gives $x$.

• Properties:

  • $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

  • $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

• Example: $\sqrt{16} = 4$

Tables: Solution Sets and Intervals

Classification of Solution Sets

Tables in the material classify solution sets using interval notation and set brackets.

Symbol	Meaning
{ }	Set notation, listing specific solutions
( )	Open interval (endpoints not included)
[ ]	Closed interval (endpoints included)
$(-\infty, a)$	All real numbers less than $a$
$(b, \infty)$	All real numbers greater than $b$

Additional info: The tables and number lines are used to visually and symbolically represent solution sets for equations and inequalities, a key skill in College Algebra.

Summary

This review covers essential topics from Chapter 1 of College Algebra, including solving equations and inequalities, graphing polynomial and quadratic functions, and representing solutions using interval notation and number lines. Mastery of these concepts is foundational for success in further algebraic studies.