College Algebra Final Exam Study Guide: Key Concepts and Examples

Absolute Value Inequalities and Equations

Solving Absolute Value Inequalities

Absolute value inequalities require isolating the absolute value expression and then rewriting as a compound inequality.

• Key Point 1: To solve $|u| < a$, rewrite as $-a < u < a$.

• Key Point 2: For $|u| > a$, rewrite as $u < -a$ or $u > a$.

• Interval Notation: Use parentheses for strict inequalities (< or >), brackets for inclusive (≤ or ≥).

• Example: Solve $|5x - 5| < 10$ Step 1: $|x - 1| < 2$ Step 2: $-2 < x - 1 < 2$ Step 3: $-1 < x < 3$ Answer: $(-1, 3)$

• Example: Solve $3|5 - \frac{3}{2}x| + 12 = 33$ Step 1: $|5 - \frac{3}{2}x| = 7$ Step 2: $5 - \frac{3}{2}x = 7$ or $5 - \frac{3}{2}x = -7$ Step 3: $x = -\frac{4}{3}$ or $x = 8$ Solution set: $\left\{ -\frac{4}{3},\ 8 \right\}$

Solving Linear Inequalities

• Key Point: Distribute, combine like terms, and isolate the variable.

• Example: $2(3 - 4x) \geq 8 - 6x$ Solution: $x \leq -1$ Interval notation: $(-\infty, -1]$

Slope, y-Intercept, and Graphing Lines

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Key Point 1: The slope $m$ represents the rate of change; $b$ is where the line crosses the y-axis.

• Example: $y = -4x + 1$ Slope: $-4$ y-intercept: $1$Graph starts at$(0, 1)$ and moves down 4, right 1 for another point.

• Example: Rewrite $6x - 2y - 8 = 0$ in slope-intercept form: $y = 3x - 4$ Slope: $3$, y-intercept:$-4$

Finding Slope from Two Points

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

• Example: Points $(2, -5)$ and $(2, -9)$ $m = \frac{-9 - (-5)}{2 - 2} = \frac{-4}{0}$ (undefined; vertical line)

Logarithms

Solving Logarithmic Equations

Logarithmic equations often require combining logs and converting to exponential form.

• Key Point 1: The domain of $\log_b(x)$ is $x > 0$.

• Key Point 2: Combine logs using $\log_b(M) + \log_b(N) = \log_b(MN)$.

• Example: $\log_{13}(x) + \log_{13}(12x - 1) = 1$ Domain: $x > 0$, $12x - 1 > 0 \implies x > \frac{1}{12}$ Combine: $\log_{13}(x(12x - 1)) = 1$ Exponential: $x(12x - 1) = 13$ Solve: $x = \frac{13}{12}$ (since $x = -1$ not in domain)

• Example: $\log_2(x + 5) = 4$ $x + 5 = 2^4 = 16 \implies x = 11$

Domain of Logarithmic Functions

• Key Point: The argument of the log must be positive.

• Example: $f(x) = \log_8(x - 8)$ $x - 8 > 0 \implies x > 8$ Domain: $(8, \infty)$

Exponential Equations and Rational Exponents

Solving Exponential Equations

• Key Point: Express both sides with the same base, then set exponents equal.

• Example: $8^{3x - 1} = 64$ $8 = 2^3$, $64 = 2^6$ $2^{9x - 3} = 2^6 \implies 9x - 3 = 6 \implies x = 1$

Solving Equations with Rational Exponents

• Key Point: Undo the exponent by raising both sides to the reciprocal power.

• Example: $(x - 4)^{2/3} = 64$ $x - 4 = 64^{3/2} = 512$ $x = 516$

Complex Solutions

• Key Point: If a square equals a negative, solutions are complex.

• Example: $(x + 9)^2 = -16$ $x + 9 = \pm 4i \implies x = -9 \pm 4i$

Compound Interest

Compound Interest Formulas

• Key Point 1: $A = P\left(1 + \frac{r}{n}\right)^{nt}$ for compounding $n$ times per year.

• Key Point 2: $A = Pe^{rt}$ for continuous compounding.

• Example: $P = 25000$, $r = 0.065$, $t = 6$

  • Semiannually ($n = 2$): $A = 25000(1 + 0.065/2)^{12} = 36719.66$

  • Quarterly ($n = 4$): $A = 25000(1 + 0.065/4)^{24} = 36795.31$

  • Monthly ($n = 12$): $A = 25000(1 + 0.065/12)^{72} = 36846.40$

  • Continuously: $A = 25000e^{0.065 \times 6} = 36945.37$

• Money answers: Round to the nearest cent.

Complex Numbers in Standard Form

Writing Complex Numbers as $a + bi$

• Key Point: To remove $i$ from the denominator, multiply numerator and denominator by the conjugate.

• Example: $\frac{8}{9 - i}$ Multiply by $9 + i$: Numerator: $8(9 + i) = 72 + 8i$ Denominator: $(9 - i)(9 + i) = 81 - (-1) = 82$ Final: $\frac{36}{41} + \frac{4}{41}i$

Functions and Composition

Function Composition

• Key Point: $(f \circ g)(x) = f(g(x))$; always work inside out.

• Example: $f(x) = 4x$, $g(x) = x + 2$ $(f \circ g)(x) = 4(x + 2) = 4x + 8$ $(g \circ f)(x) = 4x + 2$ $(f \circ g)(2) = 16$ $(g \circ f)(2) = 10$

Reading Graphs

Domain and Range

• Key Point: Domain is all $x$-values; range is all $y$-values covered by the graph.

• Notation: Use brackets $[\ ]$ for included endpoints, parentheses $(\ )$ for excluded or infinity.

Increasing, Decreasing, and Constant Intervals

• Increasing: Graph rises left to right.

• Decreasing: Graph falls left to right.

• Constant: Graph is flat.

Parabola Graphs

• Vertex: Lowest (upward) or highest (downward) point.

• Axis of symmetry: Vertical line through the vertex.

• x-intercepts: Where $y = 0$.

• y-intercept: Where $x = 0$.

• Example: $f(x) = (x + 1)^2 - 9$ Vertex: $(-1, -9)$ Axis: $x = -1$ Domain: $(-\infty, \infty)$ Range: $[-9, \infty)$ x-intercepts: $x = 2$ or $x = -4$ y-intercept: $y = -8$

Systems of Equations and Inequalities

Solving Systems of Equations

• Key Point: Use substitution or elimination. If equations are equivalent, infinitely many solutions.

• Example: $x = 8y + 1$, $-3x + 24y = -3$ Substitute and simplify: $-3 = -3$ (always true) Conclusion: Infinitely many solutions (same line)

Graphing Systems of Inequalities

• Key Point: The solution is the overlap of shaded regions.

• Example: $x^2 + y^2 \geq 25$ (circle, shade outside/on), $x + y > 3$ (line, shade above)

Exponential and Quadratic Graphs

Exponential and Logarithmic Graphs

• Key Point: $f(x) = a^x$ is exponential; $g(x) = \log_a x$ is logarithmic. They are inverses.

• Example: $f(x) = 4^x$ passes through $(0, 1)$, horizontal asymptote $y = 0$ $g(x) = \log_4 x$ passes through $(1, 0)$, vertical asymptote $x = 0$ Their graphs reflect across $y = x$

Quadratic Graphs

• Vertex form: $y = a(x - h)^2 + k$; vertex at $(h, k)$, axis $x = h$.

• Domain: $(-\infty, \infty)$ for all quadratics.

• Range: $[k, \infty)$ if $a > 0$; $(-\infty, k]$ if $a < 0$.

Formula Bank

Formula / Rule	Use it for
$A = P(1 + \frac{r}{n})^{nt}$	Compound interest
$A = Pe^{rt}$	Continuous compounding
$\log_b(M) + \log_b(N) = \log_b(MN)$	Combine logs with same base
$\log_b(x) = y \iff b^y = x$	Rewrite logarithms as exponentials
$m = \frac{y_2 - y_1}{x_2 - x_1}$	Slope from two points
$|u| < a \iff -a < u < a$	Absolute value inequality
$|u| = a \iff u = a$ or $u = -a$	Absolute value equation
Vertex form $y = a(x-h)^2 + k$	Vertex is $(h, k)$, axis is $x = h$

Answer Formats

• Interval notation: $(-1, 3]$, $(8, \infty)$, $(-\infty, -1]$

• Set notation: $\{11\}$, $\{13/12\}$, $\{-4/3, 8\}$

• Slope-intercept form: $y = mx + b$

• Complex number form: $a + bi$

• Ordered pair solution: $(x, y)$

• Money: 36719.66