뒤로Polynomial and Rational Inequalities: Study Notes for College Algebra
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Polynomial and Rational Functions
Polynomial and Rational Inequalities
This section explores the methods for solving inequalities involving polynomial and rational functions. These concepts are fundamental in College Algebra, as they provide tools for analyzing the behavior of functions and their solution sets.
Definition of a Polynomial Inequality
Polynomial Inequality: An inequality that can be written in the form , , , or , where is a polynomial function.
Examples include: or .
Procedure for Solving Polynomial Inequalities
To solve a polynomial inequality, follow these steps:
Express the inequality in the form or , where is a polynomial function.
Solve the equation . The real solutions are called boundary points.
Locate boundary points on a number line, dividing the line into intervals.
Choose a test value within each interval and evaluate at that number:
If is positive, then for all in the interval.
If is negative, then for all in the interval.
Write the solution set by selecting the intervals that satisfy the original inequality.
Note: If the inequality involves or , include the boundary points in the solution set.
Example: Solving a Polynomial Inequality
Step 1: Express the inequality in the form or .
Step 2: Solve to find boundary points.
Step 3: Locate boundary points on a number line, dividing it into intervals.
Step 4: Choose a test value in each interval and evaluate .
Step 5: Write the solution set, selecting intervals where the inequality holds.
Example: If for in and , then the solution set is .
Definition of a Rational Inequality
Rational Inequality: An inequality that can be written in the form , , , or , where and are polynomial functions and .
Examples include: .
Procedure for Solving Rational Inequalities
Express the inequality so that one side is zero and the other side is a single quotient.
Set the numerator and denominator equal to zero to find boundary points.
Locate boundary points on a number line, dividing it into intervals.
Choose a test value in each interval and evaluate the rational function.
Write the solution set by selecting intervals where the inequality holds.
Note: Exclude points where the denominator is zero from the solution set.
Example: Solving a Rational Inequality
Step 1: Express the inequality as or .
Step 2: Set and to find boundary points.
Step 3: Locate boundary points on a number line, dividing it into intervals.
Step 4: Choose a test value in each interval and evaluate .
Step 5: Write the solution set, selecting intervals where the inequality holds.
Example: If the solution set is , graph these intervals on a number line.
Application: Position Function for a Free-Falling Object
The height of an object falling or projected vertically near Earth's surface is given by the position function:
: Initial velocity (feet per second)
: Initial height (feet)
: Time in seconds
Example: Application of Polynomial Inequality
An object is propelled straight up from ground level with an initial velocity of 80 feet per second. Its height at time is modeled by:
To find when the object is more than 64 feet above the ground, solve:
Step 1: Express the inequality in the form .
Step 2: Solve to find boundary values.
Step 3: Test intervals between boundary values.
Step 4: The solution is (object is above 64 feet between 1 and 4 seconds).
Example: The object will be more than 64 feet above the ground between 1 and 4 seconds, excluding and .
