Equations and Inequalities: College Algebra Study Guide

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations involves isolating the variable on one side of the equation.

• Definition: A linear equation in one variable has the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Steps to Solve:

  1. Combine like terms on each side.

  2. Isolate the variable using addition/subtraction.

  3. Solve for the variable using multiplication/division.

• Example: Solve $2x + 3 = 15$

  • Subtract 3: $2x = 12$

  • Divide by 2: $x = 6$

Solving Rational Equations

Rational equations contain fractions with polynomials in the numerator and denominator. To solve, clear denominators by multiplying both sides by the least common denominator (LCD).

• Definition: A rational equation is an equation containing rational expressions.

• Steps to Solve:

  1. Find the LCD of all denominators.

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

  3. Solve the resulting equation.

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

• Example: Solve $\frac{x}{2} = \frac{3}{4}$

  • Multiply both sides by 4: $2x = 3$

  • Divide by 2: $x = \frac{3}{2}$

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Definition: A quadratic equation is a second-degree equation in the form $ax^2 + bx + c = 0$.

• Quadratic Formula:

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

• Example: Solve $x^2 - 4 = 0$

  • Factor: $(x - 2)(x + 2) = 0$

  • Solutions: $x = 2$, $x = -2$

Solving Radical Equations

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

• Definition: A radical equation contains a variable within a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$.

• Steps to Solve:

  1. Isolate the radical expression.

  2. Raise both sides to the power that eliminates the radical.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: Solve $\sqrt{x} = 5$

  • Square both sides: $x = 25$

Solving Absolute Value Equations and Inequalities

Absolute value equations and inequalities involve expressions within $|\ |$. The solution often splits into two cases, one positive and one negative.

• Definition: The absolute value of $x$, $|x|$, is the distance from $x$ to 0 on the number line.

• Equation: $|x| = a$ has solutions $x = a$ and $x = -a$.

• Inequality: $|x| < a$ means $-a < x < a$; $|x| > a$ means $x < -a$ or $x > a$.

• Example: Solve $|x - 3| = 5$

  • Case 1: $x - 3 = 5 \Rightarrow x = 8$

  • Case 2: $x - 3 = -5 \Rightarrow x = -2$

Solving and Graphing Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols ($<, >, \leq, \geq$). Solutions are often expressed in interval notation.

• Definition: A linear inequality is an inequality involving a linear expression.

• Steps to Solve:

  1. Isolate the variable.

  2. If multiplying or dividing by a negative, reverse the inequality sign.

  3. Express the solution in interval notation.

• Example: Solve $2x - 5 < 9$

  • Add 5: $2x < 14$

  • Divide by 2: $x < 7$

  • Interval notation: $(-\infty, 7)$

Applications: Word Problems and Modeling

Equations and inequalities are used to model and solve real-world problems, such as geometry, finance, and measurement.

• Example: The length and width of a rectangle are related by an equation. If the area is known, set up and solve an equation to find the dimensions.

• Example: A rectangular piece of metal is used to form a box. Use the given dimensions and volume to write and solve an equation.

Interval Notation

Interval notation is a way to describe sets of numbers, especially solutions to inequalities.

• Open Interval: $(a, b)$ means all numbers between $a$ and $b$, not including $a$ or $b$.

• Closed Interval: $[a, b]$ means all numbers between $a$ and $b$, including $a$ and $b$.

• Infinite Intervals: $(-\infty, a)$ or $(b, \infty)$

• Example: $x > 3$ is written as $(3, \infty)$

Summary Table: Types of Equations and Solution Methods

Type of Equation	General Form	Solution Method	Example
Linear	$ax + b = c$	Isolate variable	$2x + 3 = 7$
Quadratic	$ax^2 + bx + c = 0$	Factoring, quadratic formula	$x^2 - 4 = 0$
Rational	$\frac{ax + b}{cx + d} = e$	Clear denominators	$\frac{x}{2} = 3$
Radical	$\sqrt{x} = a$	Isolate radical, raise to power	$\sqrt{x} = 5$
Absolute Value	$|x| = a$	Split into two cases	$|x - 3| = 5$
Inequality	$ax + b < c$	Isolate variable, interval notation	$2x - 5 < 9$

Additional info:

• This study guide covers topics from College Algebra Chapter 1: Equations and Inequalities, including linear, quadratic, rational, radical, and absolute value equations, as well as inequalities and applications.

• Examples and solution methods are expanded for clarity and completeness.