College Algebra: Rational Equations, Inequalities, and Functions Study Guide

Rational Equations and Restrictions

Identifying Restrictions in Rational Equations

When working with rational equations, it is important to determine the values of the variable that make any denominator zero, as these are not allowed in the domain.

• Restriction: A value that makes the denominator of a rational expression equal to zero.

• How to Find: Set each denominator equal to zero and solve for the variable.

• Example: For $\frac{2}{x+3} - \frac{1}{x-2} = \frac{x+1}{x^2 + x - 6}$, factor the denominator: $x^2 + x - 6 = (x+3)(x-2)$. The restrictions are $x \neq -3$, $x \neq 2$.

Clearing Fractions in Rational Equations

Multiplying by the Least Common Denominator (LCD)

To solve rational equations, multiply both sides by the LCD to eliminate denominators.

• LCD: The least common denominator is the smallest expression that is a multiple of all denominators in the equation.

• Example: $\frac{4}{x+1} + \frac{1}{x+2} = \frac{5x+1}{(x+1)(x+2)}$ Multiply both sides by $(x+1)(x+2)$ to clear denominators: $4(x+2) + 1(x+1) = 5x+1$

Solving Rational Equations

Step-by-Step Solutions

After clearing denominators, solve the resulting linear or quadratic equation.

• Check for Extraneous Solutions: Always check that your solutions do not make any denominator zero in the original equation.

• Example: $8x\left(\frac{2}{x}\right) + 8x(1) = 8x\left(\frac{3}{4x}\right) + 8x\left(\frac{15}{8}\right)$ Solve for $x$ and check restrictions.

Factoring and Solving Quadratic Equations

Factoring Quadratics

Quadratic equations can often be solved by factoring.

• Standard Form: $ax^2 + bx + c = 0$

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

• Example: $x^2 - 6x - 16 = 0$ Factor: $(x-8)(x+2) = 0$ Solutions: $x = 8$, $x = -2$

Solving and Graphing Inequalities

Linear Inequalities

To solve inequalities, isolate the variable and express the solution in interval notation. Graph the solution on a number line.

• Example: $-6x \leq 18$ $x \geq -3$ Interval Notation: $[-3, \infty)$

Compound Inequalities

• Example: $-8 < 2x - 2 \leq 8$ Solve for $x$ to get $-3 < x \leq 5$ Interval Notation: $(-3, 5]$

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Set up two equations: one for the positive case and one for the negative case.

• Example: $|5x-2| = 13$ $5x-2 = 13$ or $5x-2 = -13$ Solutions: $x = 3$, $x = -\frac{11}{5}$

Solving Absolute Value Inequalities

• Example: $|2x-1| < 3$ $-3 < 2x-1 < 3$ $-1 < x < 2$ Interval Notation: $(-1, 2)$

• Example: $|x-4| \geq 6$ $x-4 \geq 6$ or $x-4 \leq -6$ $x \geq 10$ or $x \leq -2$ Interval Notation: $(-\infty, -2] \cup [10, \infty)$

Functions, Domain, and Range

Determining if a Relation is a Function

A relation is a function if every input (x-value) corresponds to exactly one output (y-value).

• Vertical Line Test: If any vertical line crosses the graph more than once, the relation is not a function.

• Example: The set $\{(3,2), (4,-1), (5,3)\}$ is a function because all x-values are different.

• Example: The set $\{(-2,3), (3,-1), (-2,2), (3,4)\}$ is not a function because $-2$ and $3$ repeat as x-values.

Domain and Range

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: For $\{(3,2), (4,-1), (5,3)\}$, Domain: $\{3,4,5\}$, Range: $\{2,-1,3\}$

Graphing and Interpreting Functions

Reading Graphs

• Domain: Read the graph from left to right.

• Range: Read the graph from bottom to top.

• x-intercepts: Points where the graph crosses the x-axis.

• y-intercept: Point where the graph crosses the y-axis.

• Function Values: $f(a)$ is the y-value when $x=a$.

• Example: If $f(3) = 5$ and $f(-4) = 4$, these are the function values at $x=3$ and $x=-4$.

Summary Table: Key Concepts

Concept	Definition	Example
Restriction	Value making denominator zero	$x \neq -3$, $x \neq 2$
Function	Each input has one output	$\{(1,2), (2,3)\}$
Domain	All possible x-values	$\{1,2,3\}$
Range	All possible y-values	$\{2,3,4\}$
Interval Notation	Describes a set of numbers between endpoints	$[a, b)$
Absolute Value Equation	Equation involving $|x|$	$|x-4| = 6$