College Algebra: Solving Equations, Simplifying Expressions, and Solving Inequalities

Solving Linear Equations

Introduction

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations is a foundational skill in algebra, involving isolating the variable to find its value.

• Definition: A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Key Steps:

  1. Combine like terms on each side.

  2. Isolate the variable by adding or subtracting terms.

  3. Solve for the variable by dividing or multiplying.

• Example: Solve $9x - 5 = 7 + 4x$

  • Subtract $4x$ from both sides: $9x - 4x - 5 = 7$

  • $5x - 5 = 7$

  • Add $5$to both sides: $5x = 12$

  • Divide by $5$:$x = \frac{12}{5}$

Solving Quadratic Equations

Introduction

Quadratic equations are equations where the variable is squared. There are several methods to solve them, including factoring, using the square root property, completing the square, and the quadratic formula.

• Definition: A quadratic equation has the form $ax^2 + bx + c = 0$.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

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

• Completing the Square: Transform the equation into $(x + d)^2 = e$ and solve for $x$.

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

• Example (Factoring): Solve $x^2 + 7x + 6 = 0$

  • Factor: $(x + 6)(x + 1) = 0$

  • Set each factor to zero: $x + 6 = 0$ or $x + 1 = 0$

  • Solutions: $x = -6$, $x = -1$

• Example (Quadratic Formula): Solve $x^2 + 2x + 26 = 0$

  • $a = 1$, $b = 2$, $c = 26$

  • $x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 26}}{2 \cdot 1}$

  • $x = \frac{-2 \pm \sqrt{4 - 104}}{2}$

  • $x = \frac{-2 \pm \sqrt{-100}}{2}$

  • $x = \frac{-2 \pm 10i}{2}$

  • $x = -1 \pm 5i$

Simplifying Algebraic and Complex Expressions

Introduction

Simplifying expressions involves combining like terms, applying arithmetic operations, and using properties of real and complex numbers.

• Complex Numbers: Numbers of the form $a + bi$, where $i = \sqrt{-1}$.

• Key Properties:

  • $(a + bi) + (c + di) = (a + c) + (b + d)i$

  • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Example: Simplify $(6 + 5i) - (8 - 6i)$

  • $(6 + 5i) - (8 - 6i) = 6 + 5i - 8 + 6i = (6 - 8) + (5i + 6i) = -2 + 11i$

• Example: Simplify $\sqrt{8} - \sqrt{3} - \sqrt{17}$

  • $\sqrt{8} = 2\sqrt{2}$

  • So, $2\sqrt{2} - \sqrt{3} - \sqrt{17}$

Solving Inequalities and Interval Notation

Introduction

Inequalities express a range of possible values for a variable. Solutions are often represented using interval notation and can be graphed on a number line.

• Definition: An inequality is a statement that compares two expressions using symbols such as $<$, $>$, $\leq$, or $\geq$.

• Interval Notation: Describes the set of solutions using parentheses and brackets. For example, $(-\infty, 7]$ means all real numbers less than or equal to $7$.

• Example: Solve $6x - 5 \leq 8$ or $6x - 5 \geq 13$

  • $6x - 5 \leq 8 \implies 6x \leq 13 \implies x \leq \frac{13}{6}$

  • $6x - 5 \geq 13 \implies 6x \geq 18 \implies x \geq 3$

  • Interval notation: $(-\infty, \frac{13}{6}] \cup [3, \infty)$

• Graphing: Solutions can be represented on a number line, with closed circles for inclusive endpoints and open circles for exclusive endpoints.

Summary Table: Methods for Solving Quadratic Equations

Method	Form of Equation	Key Steps	Example
Factoring	$ax^2 + bx + c = 0$	Express as product of binomials, set each to zero	$(x + 6)(x + 1) = 0$
Square Root Property	$x^2 = a$	Take square root of both sides	$x^2 = 25 \implies x = \pm 5$
Completing the Square	$ax^2 + bx + c = 0$	Rewrite as $(x + d)^2 = e$	$x^2 + 7x + 4 = 0$
Quadratic Formula	$ax^2 + bx + c = 0$	Apply $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 + 2x + 26 = 0$

Additional info:

• Some problems involve complex solutions, indicated by the presence of $i$ in the answer.

• Interval notation and graphing are essential for expressing solutions to inequalities.

• Problems cover a range of College Algebra topics, including linear and quadratic equations, complex numbers, and inequalities.