College Algebra: Equations, Graphs, and Applications – Comprehensive Study Notes

Equations & Inequalities

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The general form is ax + b = c.

• Solving Linear Equations: Isolate the variable using inverse operations (addition, subtraction, multiplication, division).

• Checking Solutions: Substitute the solution back into the original equation to verify correctness.

• Conditional, Identity, and Inconsistent Equations:

  • Conditional: True for specific values.

  • Identity: True for all values of the variable.

  • Inconsistent: No solution exists.

• Example: Solve $2x + 3 = 7$ Subtract 3: $2x = 4$ Divide by 2: $x = 2$

Equations with Fractions and Decimals

Equations may include fractions or decimals. Clear fractions by multiplying both sides by the least common denominator (LCD).

• Example: Solve $\frac{x}{3} + 2 = 5$ Subtract 2: $\frac{x}{3} = 3$ Multiply by 3: $x = 9$

Equations with Variables in Denominators

Some equations have variables in the denominator. Identify values that make the denominator zero (restrictions).

• Example: $\frac{1}{x-2} = 3$ Restriction: $x \neq 2$

Quadratic Equations

Quadratic equations have the form $ax^2 + bx + c = 0$. Solutions can be found by factoring, completing the square, or using the quadratic formula.

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

• Square Root Property: If $x^2 = k$, then $x = \pm \sqrt{k}$

• Example: Solve $x^2 = 9$ $x = \pm 3$

Graphs of Equations

Plotting Points

Points are plotted on the rectangular coordinate system (Cartesian plane) using ordered pairs (x, y).

• Example: Plot (1, -5): Move 1 unit right on the x-axis, 5 units down on the y-axis.

Graphing Linear Equations

Linear equations graph as straight lines. The slope-intercept form is $y = mx + b$, where m is the slope and b is the y-intercept.

• Finding Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: $y = 2x + 3$ has slope 2 and y-intercept 3.

Graphing Quadratic Equations

Quadratic equations graph as parabolas. The vertex form is $y = a(x-h)^2 + k$.

• Vertex: The point (h, k) is the vertex of the parabola.

• Example: $y = x^2$ has vertex at (0, 0).

Interpreting Graphs

Graphs can be used to determine intercepts, slope, and the general behavior of equations.

• x-intercept: Where the graph crosses the x-axis ($y=0$).

• y-intercept: Where the graph crosses the y-axis ($x=0$).

Functions

Definition of a Function

A function is a relation in which each input (x-value) has exactly one output (y-value).

• Function Notation: $f(x)$ represents the output when input is $x$.

• Example: $f(x) = x^2$

Evaluating Functions

To evaluate a function, substitute the given value for x.

• Example: If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$

Review of Algebra

Order of Operations

Follow the order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Example: $2 + 3 \times 4 = 2 + 12 = 14$

Properties of Real Numbers

Real numbers follow properties such as commutative, associative, and distributive laws.

• Commutative: $a + b = b + a$

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

• Distributive: $a(b + c) = ab + ac$

Operations with Radicals

Radicals are expressions involving roots. Simplify by factoring and using properties of exponents.

• Example: $\sqrt{16} = 4$

• Multiplying Radicals: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

Applications

Word Problems

Translate real-world scenarios into algebraic equations to solve for unknowns.

• Example: If a car rental costs $25 per day plus $0.15 per mile, and the total cost is $62, set up the equation: $25 + 0.15m = 62$

• Solving: $0.15m = 37 \Rightarrow m = \frac{37}{0.15}$

Interest Problems

Simple interest is calculated using $I = Prt$, where P is principal, r is rate, and t is time.

• Example: Invest $2000$at $5\%$ for $1$year:$I = 2000 \times 0.05 \times 1 = 100$

Tables: Ordered Pairs and Graphs

Purpose: Comparing Ordered Pairs and Their Graphs

x	y
-2	4
-1	1
0	0
1	1
2	4

This table represents ordered pairs for $y = x^2$. The graph is a parabola opening upwards.

Summary Table: Types of Equations

Type	Description	Example
Linear	Degree 1, straight line graph	$y = 2x + 3$
Quadratic	Degree 2, parabola graph	$y = x^2$
Identity	True for all values	$x = x$
Conditional	True for specific values	$x + 2 = 5$
Inconsistent	No solution	$x + 2 = x + 3$

Additional info:

• Some questions involve graph interpretation, function evaluation, and application problems, all central to College Algebra.

• Quadratic equations are solved by multiple methods, including factoring, square root property, and quadratic formula.

• Radical operations and simplification are included, as well as word problems involving rates, interest, and geometry.