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 , rewrite as .

• Key Point 2: For , rewrite as or .

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

• Example: Solve Step 1: Step 2: Step 3: Answer:

• Example: Solve Step 1: Step 2: or Step 3: or Solution set:

Solving Linear Inequalities

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

• Example: Solution: Interval notation:

Slope, y-Intercept, and Graphing Lines

Slope-Intercept Form

The slope-intercept form of a line is , where is the slope and is the y-intercept.

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

• Example: Slope: y-intercept: $1(0, 1)$ and moves down 4, right 1 for another point.

• Example: Rewrite in slope-intercept form: Slope: $3-4$

Finding Slope from Two Points

• Formula:

• Example: Points and (undefined; vertical line)

Logarithms

Solving Logarithmic Equations

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

• Key Point 1: The domain of is .

• Key Point 2: Combine logs using .

• Example: Domain: , Combine: Exponential: Solve: (since not in domain)

• Example:

Domain of Logarithmic Functions

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

• Example: Domain:

Exponential Equations and Rational Exponents

Solving Exponential Equations

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

• Example: ,

Solving Equations with Rational Exponents

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

• Example:

Complex Solutions

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

• Example:

Compound Interest

Compound Interest Formulas

• Key Point 1: for compounding times per year.

• Key Point 2: for continuous compounding.

• Example: , ,

  • Semiannually ():

  • Quarterly ():

  • Monthly ():

  • Continuously:

• Money answers: Round to the nearest cent.

Complex Numbers in Standard Form

Writing Complex Numbers as

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

• Example: Multiply by : Numerator: Denominator: Final:

Functions and Composition

Function Composition

• Key Point: ; always work inside out.

• Example: ,

Reading Graphs

Domain and Range

• Key Point: Domain is all -values; range is all -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-intercept: Where .

• Example: Vertex: Axis: Domain: Range: x-intercepts: or y-intercept:

Systems of Equations and Inequalities

Solving Systems of Equations

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

• Example: , Substitute and simplify: (always true) Conclusion: Infinitely many solutions (same line)

Graphing Systems of Inequalities

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

• Example: (circle, shade outside/on), (line, shade above)

Exponential and Quadratic Graphs

Exponential and Logarithmic Graphs

• Key Point: is exponential; is logarithmic. They are inverses.

• Example: passes through , horizontal asymptote passes through , vertical asymptote Their graphs reflect across

Quadratic Graphs

• Vertex form: ; vertex at , axis .

• Domain: for all quadratics.

• Range: if ; if .

Formula Bank

Answer Formats

• Interval notation: , ,

• Set notation: , ,

• Slope-intercept form:

• Complex number form:

• Ordered pair solution:

• Money: 36719.66