Equations and Inequalities

Solving Rational Equations Leading to Quadratic Equations

Rational equations are equations involving fractions with polynomials in the numerator and denominator. Sometimes, solving these equations results in a quadratic equation.

• Key Point: Clear denominators by multiplying both sides by the least common denominator (LCD).

• Key Point: Rearrange the resulting equation into standard quadratic form: .

• Example: Solve . Multiply both sides by , expand, and solve the resulting quadratic.

Solving Equations Involving Radicals

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

• Key Point: Isolate the radical expression before eliminating it.

• Key Point: Check for extraneous solutions caused by squaring both sides.

• Example: Solve . Square both sides and solve the resulting equation.

Solving Equations with Rational Exponents

Equations with rational exponents can be solved by rewriting the exponents as roots and powers, then isolating the variable.

• Key Point: Rewrite as .

• Key Point: Raise both sides to the reciprocal power to eliminate the exponent.

• Example: Solve . Raise both sides to the power: .

Solving Linear Inequalities

Linear inequalities involve expressions of degree one. The solution is often a range of values.

• Key Point: Solve as you would a linear equation, but reverse the inequality sign if multiplying/dividing by a negative.

• Example: Solve . Add 5, divide by 2: .

Solving Quadratic Inequalities

Quadratic inequalities involve expressions of degree two. Solutions are often intervals.

• Key Point: Set the inequality to zero and solve the corresponding quadratic equation.

• Key Point: Test intervals between roots to determine where the inequality holds.

• Example: Solve . Roots are and . Test intervals: , , .

Solving Rational Inequalities

Rational inequalities involve fractions with polynomials. Solutions are found by determining where the expression is positive or negative.

• Key Point: Find zeros of numerator and denominator.

• Key Point: Create a sign chart to test intervals.

• Example: Solve . Zeros at (numerator) and (denominator).

Solving Absolute Value Equations

Absolute value equations require considering both the positive and negative cases.

• Key Point: leads to or .

• Example: Solve . Solutions: or ; or .

Solving Absolute Value Inequalities

Absolute value inequalities are solved by splitting into two cases, depending on the direction of the inequality.

• Key Point: leads to ; leads to or .

• Example: Solve . Solution: ; .

Analytic Geometry

Distance and Midpoint Between Two Points

Given two points and , the distance and midpoint formulas are fundamental in analytic geometry.

• Distance Formula:

• Midpoint Formula:

• Example: Points and : ; .

Graphing Equations by Finding Ordered Pair Solutions

To graph an equation, select values for one variable and solve for the other to obtain ordered pairs.

• Key Point: Choose several values for , solve for , and plot the resulting pairs.

• Example: For , gives ; gives .

Identifying the Center and Radius of a Circle from Its Equation

The standard form of a circle's equation is , where is the center and is the radius.

• Key Point: Compare the given equation to the standard form to identify center and radius.

• Example: has center and radius .

Graphing a Circle Given Its Equation or Center and Radius

To graph a circle, plot the center and use the radius to draw all points equidistant from the center.

• Key Point: Use the standard form to identify center and radius.

• Example: ; center , radius $3$.

Graphs and Functions

Evaluating Functions

To evaluate a function, substitute the input value into the function's formula.

• Key Point: If , then .

• Example: , .

Determining Whether a Relation Defines a Function

A function assigns exactly one output to each input. A relation is a function if no input value is paired with more than one output.

• Key Point: Use the vertical line test on a graph: if any vertical line crosses the graph more than once, it is not a function.

• Example: is a function; is not (circle).

Graphing Linear Functions and Lines When Equations Are Given

Linear functions have the form , where is the slope and is the y-intercept.

• Key Point: Plot the y-intercept, use the slope to find another point, and draw the line.

• Example: ; y-intercept at , slope means down 2, right 1.