College Algebra Practice Test 1: Comprehensive Study Notes

Equations and Inequalities

Solving Rational Equations

Rational equations are equations involving fractions whose numerators and/or denominators contain algebraic expressions. To solve them, find a common denominator and clear the fractions.

• Key Steps:

  1. Identify the least common denominator (LCD).

  2. Multiply both sides of the equation by the LCD to eliminate denominators.

  3. Solve the resulting polynomial equation.

  4. Check for extraneous solutions by substituting back into the original equation.

• Example: Solve $\frac{1}{x+3} = \frac{2}{x-1} + \frac{2x+2}{x^2+2x-15}$

  • Factor denominators and find LCD: $x^2+2x-15 = (x+5)(x-3)$

  • Multiply both sides by LCD and solve for $x$.

Solving Linear and Quadratic Equations

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

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

• Example: Solve $3x^2 + 4x - 4 = 0$ using the quadratic formula.

Solving Radical Equations

Radical equations contain variables within a root. Isolate the radical and raise both sides to the appropriate power to eliminate the root.

• Example: Solve $\sqrt{x^2 + 1} = \sqrt{2x + 5}$

Solving Absolute Value Equations and Inequalities

Absolute value equations and inequalities require considering both the positive and negative cases.

• General Principle: $|A| = B$ implies $A = B$ or $A = -B$

• Example: $|2x + 3| = |2x - 5|$

Solving Inequalities

Inequalities can be linear, quadratic, or rational. Solutions are often expressed in interval notation.

• Example: Solve $\frac{x^2 - 4x + 3}{x^2 - x - 12} \leq 0$

• Interval Notation: Use brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints.

Functions and Their Graphs

Identifying Functions

A relation is a function if each input (domain value) corresponds to exactly one output (range value).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Example: Given the set of ordered pairs {(3, -1), (5, 0), (2, 6), (-1, -1)}, determine if it is a function.

• Mapping Diagram: If any input maps to more than one output, it is not a function.

• Equation Example: $x^2 + y^2 = 100$ is not a function of $x$ because for some $x$ values, there are two corresponding $y$ values.

Domain of Functions

The domain of a function is the set of all input values for which the function is defined.

• Rational Functions: Exclude values that make the denominator zero.

• Radical Functions: For even roots, the radicand must be non-negative.

• Examples:

  • $f(x) = \frac{8x}{x^2 - 4x + 1}$

  • $g(x) = \sqrt{x^2 - 2x - 3}$

Linear and Quadratic Functions

Distance, Midpoint, and Equation of a Line

Given two points, you can find the midpoint, distance, and the equation of the line passing through them.

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Equation of a Line: $y - y_1 = m(x - x_1)$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Find the midpoint, distance, and equation of the line through $(-2, -5)$ and $(4, -1)$.

Standard Form of a Circle

The equation of a circle in standard form is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• To convert from general form: Complete the square for both $x$ and $y$ terms.

• Example: $x^2 + y^2 + 2x - 10y + 1 = 0$

Complex Numbers

Simplifying Complex Quotients

To simplify a quotient involving complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Example: $\frac{3i - 2}{2 - 4i}$, express in $a + bi$ form.

Powers of i

The imaginary unit $i$ is defined as $i = \sqrt{-1}$. Powers of $i$ repeat every four terms.

• Key Values:

  • $i^1 = i$

  • $i^2 = -1$

  • $i^3 = -i$

  • $i^4 = 1$

• Example: Simplify $i^{77}$

Applications and Word Problems

Investment Problems

Set up equations based on total investment and interest rates to solve for unknowns.

• Example: An executive invests $22,000, some at 7% and the rest at 6%. If the annual return is $1,426, how much is invested at each rate?

Distance, Rate, and Time Problems

Use the formula $\text{Distance} = \text{Rate} \times \text{Time}$ to solve problems involving travel.

• Example: Jake drives 120 km, returns 10 km/h faster, and saves 1 hour. Find his speeds.

Variation

Direct and Inverse Variation

Direct variation means one variable is a constant multiple of another; inverse variation means one variable is a constant divided by another.

• General Form:

  • Direct: $y = kx$

  • Inverse: $y = \frac{k}{x}$

  • Combined: $A = kB\sqrt{C}$

• Example: Find $k$ if $A$ varies directly as $B$ and inversely as $\sqrt{C}$, given $A = 8$ when $B = 5$ and $C = 100$.

Summary Table: Key Formulas and Concepts

Concept	Formula	Example
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$3x^2 + 4x - 4 = 0$
Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Between $(-2, -5)$ and $(4, -1)$
Midpoint Formula	$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Between $(-2, -5)$ and $(4, -1)$
Standard Form of Circle	$(x - h)^2 + (y - k)^2 = r^2$	Convert $x^2 + y^2 + 2x - 10y + 1 = 0$
Complex Conjugate	If $z = a + bi$, conjugate is $a - bi$	Simplify $\frac{3i - 2}{2 - 4i}$
Variation	$A = kB\sqrt{C}$	Find $k$ given $A, B, C$

Additional info:

• Some questions involve analytic geometry (distance, midpoint, circle equations), which is covered in College Algebra.

• Complex numbers and their powers are included, relevant to polynomial equations.

• All topics are directly related to College Algebra chapters listed in the prompt.