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

Rational Equations and Restrictions

Identifying Restrictions in Rational Equations

Rational equations contain variables in the denominator. Restrictions are values that make any denominator zero, which are excluded from the domain.

• Restriction: A value of the variable that makes the denominator zero.

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

• How to Find: Set each denominator equal to zero and solve for $x$.

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 multiple of all denominators.

• Example: $\frac{4}{x+10} + \frac{1}{x+2} = \frac{5x+1}{(x+10)(x+2)}$ Multiply both sides by $(x+10)(x+2)$: $4(x+2) + 1(x+10) = 5x+1$

Solving Rational Equations

Step-by-Step Solutions

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

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

• Check for Extraneous Solutions: Substitute solutions back into the original equation to ensure they do not make any denominator zero.

Factoring Quadratic Equations

Solving by Factoring

Quadratic equations can often be solved by factoring and setting each factor equal to zero.

• Example: $x^2 - 6x - 10 = 0$ Factor: $(x-2)(x+5) = 0$ Solutions: $x = 2$, $x = -5$

• Check: Substitute solutions into the original equation to verify.

Interval Notation and Graphing Solutions

Expressing Solutions Using Interval Notation

Interval notation is used to describe the set of solutions for inequalities.

• Example: $x \geq 7$ is written as $[7, \infty)$

• Graphing: Draw a number line and shade the region representing the solution set.

Solving Linear and Absolute Value Inequalities

Linear Inequalities

To solve linear inequalities, isolate the variable and express the solution in interval notation.

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

Absolute Value Inequalities

Absolute value inequalities split into two cases.

• 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: $(-\infty, -2] \cup [10, \infty)$

Solving Absolute Value Equations

Splitting into Two Cases

For $|ax+b| = c$, solve $ax+b = c$ and $ax+b = -c$.

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

Functions, Domain, and Range

Determining if a Relation is a Function

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

• Function Test: No x-value repeats.

• Domain: Set of all x-values.

• Range: Set of all y-values.

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

• Example: $\{(-2,3), (3,1), (-2,2), (3,4)\}$ Function: No (x-value -2 repeats)

Vertical Line Test

A graph represents a function if no vertical line intersects the graph at more than one point.

• Example: A circle fails the vertical line test, so $y$ is not a function of $x$.

Graphing and Interpreting Functions

Domain and Range from Graphs

Read the domain (left to right) and range (bottom to top) from the graph.

• Domain: All possible x-values.

• Range: All possible y-values.

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

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

• Example: If the graph starts at $x=0$ and continues to the right, domain is $(0, \infty)$.

• Example: If the graph starts at $y=1$ and continues upward, range is $[1, \infty)$.

Evaluating Functions

To find $f(a)$, substitute $a$ into the function.

• Example: If $f(x)$ is given, $f(3) = 5$, $f(4) = 4$.