Equations, Inequalities, and Functions: College Algebra Study Notes

Equations & Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable on one side.

• Definition: A linear equation has the form $ax + b = c$.

• Key Steps:

  1. Combine like terms.

  2. Isolate the variable using addition/subtraction.

  3. Solve for the variable using multiplication/division.

• Example: Solve $2x + 5 = 13$.

  • Subtract 5: $2x = 8$

  • Divide by 2: $x = 4$

Solving Quadratic Equations

Quadratic equations are second-degree equations, typically written as $ax^2 + bx + c = 0$. They 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 $x^2 - 5x + 6 = 0$.

  • Factor: $(x-2)(x-3) = 0$

  • Solutions: $x = 2$, $x = 3$

Solving Radical Equations

Radical equations contain variables within a root. To solve, isolate the radical and then raise both sides to the appropriate power.

• Example: Solve $\sqrt{x+3} = 5$.

  • Square both sides: $x+3 = 25$

  • Solve: $x = 22$

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and/or denominator. Clear denominators by multiplying both sides by the least common denominator (LCD).

• Example: Solve $\frac{2}{x} = 4$.

  • Multiply both sides by $x$: $2 = 4x$

  • Divide by 4: $x = \frac{1}{2}$

Functions

Definition and Notation

A function is a relation that assigns exactly one output to each input. Functions are commonly written as $f(x)$.

• Definition: A function $f$ from set $A$ to set $B$ is a rule that assigns to each element $x$ in $A$ exactly one element $f(x)$ in $B$.

• Example: $f(x) = 2x + 1$

Evaluating Functions

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

• Example: If $f(x) = x^2 - 3x + 2$, find $f(4)$.

  • $f(4) = 4^2 - 3 \times 4 + 2 = 16 - 12 + 2 = 6$

Piecewise Functions

Piecewise functions are defined by different expressions depending on the input value.

• Example:

  • $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$

Graphs of Equations

Graphing Linear and Quadratic Equations

Graphing equations helps visualize solutions and relationships. Linear equations graph as straight lines; quadratics as parabolas.

• Linear Equation: $y = mx + b$

• Quadratic Equation: $y = ax^2 + bx + c$

• Example: Graph $y = 2x + 3$ and $y = x^2 - 4$

Systems of Equations

Solving Systems by Substitution and Elimination

Systems of equations involve finding values that satisfy multiple equations simultaneously. Common methods include substitution and elimination.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Example:

  • System: $x + y = 5$, $x - y = 1$

  • Add: $2x = 6 \Rightarrow x = 3$

  • Substitute: $3 + y = 5 \Rightarrow y = 2$

Additional info:

• Some equations and steps were inferred from partial or unclear handwriting, but all topics are standard in College Algebra.

• Content covers solving equations, function evaluation, and basic graphing, which are foundational for College Algebra.