Lesson 4-6 Rational Equations and Partial Fractions

Slides:



Advertisements
Similar presentations
Rational Equations and Partial Fraction Decomposition
Advertisements

Example 1 Replace T(d) with 45 Multiply both sides by the denominator Solve for d The maximum depth the diver can go without decompression is approximately.
Rational Expressions To add or subtract rational expressions, find the least common denominator, rewrite all terms with the LCD as the new denominator,
Algebraic Fractions and Rational Equations. In this discussion, we will look at examples of simplifying Algebraic Fractions using the 4 rules of fractions.
Table of Contents First, find the least common denominator (LCD) of all fractions present. Linear Equations With Fractions: Solving algebraically Example:
Solving Rational Equations A Rational Equation is an equation that contains one or more rational expressions. The following are rational equations:
Ch 8 - Rational & Radical Functions 8.5 – Solving Rational Equations.
Lesson 4-6 Rational Equations and Partial Fractions
Chapter 6 Section 6 Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Equations with Rational Expressions Distinguish between.
Review Solve each equation or inequality b = 2, -5 Multiply every term by 12.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 7 Systems of Equations and Inequalities.
Rational Equations and Partial Fractions The purpose of this lesson is solving Rational Equations (aka:fractions) and breaking a rational expression into.
( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9
Solving Rational Equations
EXAMPLE 2 Rationalize denominators of fractions Simplify
6.1 The Fundamental Property of Rational Expressions Rational Expression – has the form: where P and Q are polynomials with Q not equal to zero. Determining.
 Inverse Variation Function – A function that can be modeled with the equation y = k/x, also xy = k; where k does not equal zero.
Simplify a rational expression
Solving Equations with Rational Expressions Distinguish between operations with rational expressions and equations with terms that are rational expressions.
Section 2.2 More about Solving Equations. Objectives Use more than one property of equality to solve equations. Simplify expressions to solve equations.
7.8 Partial Fractions. If we wanted to add something like this: We would need to find a common denominator! Now, we want to REVERSE the process! Go from.
Chapter 7 Systems of Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc Partial Fractions.
10-7 Solving Rational Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview.
Sullivan Algebra and Trigonometry: Section 4.5 Solving Polynomial and Rational Inequalities Objectives Solve Polynomial Inequalities Solve Rational Inequalities.
Holt McDougal Algebra 2 Solving Rational Equations and Inequalities Solving Rational Equations and Inequalities Holt Algebra 2Holt McDougal Algebra 2.
Math – Rational Equations 1. A rational equation is an equation that has one or more rational expressions in it. To solve, we start by multiplying.
Multi-Step Equations We must simplify each expression on the equal sign to look like a one, two, three step equation.
Section 6.4 Rational Equations
1/20/ :24 AM10.3 Multiplying and Dividing Expressions1 Simplify, Multiply and Divide Rational Expressions Section 8-2.
(x+2)(x-2).  Objective: Be able to solve equations involving rational expressions.  Strategy: Multiply by the common denominator.  NOTE: BE SURE TO.
Solving Equations Containing First, we will look at solving these problems algebraically. Here is an example that we will do together using two different.
9-6 SOLVING RATIONAL EQUATIONS & INEQUALITIES Objectives: 1) The student will be able to solve rational equations. 2) The student will be able to solve.
Aim: How do we solve rational inequalities? Do Now: 2. Find the value of x for which the fraction is undefined 3. Solve for x: HW: p.73 # 4,6,8,10,14.
Lesson 8-6: Solving Rational Equations and Inequalities.
9.1 Simplifying Rational Expressions Objectives 1. simplify rational expressions. 2. simplify complex fractions.
Sullivan PreCalculus Section 3.5 Solving Polynomial and Rational Inequalities Objectives Solve Polynomial Inequalities Solve Rational Inequalities.
A rational expression is a fraction with polynomials for the numerator and denominator. are rational expressions. For example, If x is replaced by a number.
8.5 Solving Rational Equations. 1. Factor all denominators 2. Find the LCD 3.Multiply every term on both sides of the equation by the LCD to cancel out.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © Cengage Learning. All rights reserved.
1. Add: 5 x2 – 1 + 2x x2 + 5x – 6 ANSWERS 2x2 +7x + 30
Sullivan Algebra and Trigonometry: Section 5
Sullivan Algebra and Trigonometry: Section 4.5
Essential Questions Solving Rational Equations and Inequalities
EQUATIONS & INEQUALITIES
( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9
9.6 Solving Rational Equations
College Algebra Chapter 1 Equations and Inequalities
Solving Equations by Factoring and Problem Solving
Multiplying and Dividing Rational Expressions
Section 1.2 Linear Equations and Rational Equations
Solving One-Step Equations
Warmups Simplify 1+ 2
Solving Rational Equations
Section 1.2 Linear Equations and Rational Equations
Algebra 1 Section 13.6.
Solving Linear Equations
Rational Expressions and Equations
Bell Ringer What is the restriction on rational expressions which results in extraneous solutions? 2. When solving equations with fractions,
12 Systems of Linear Equations and Inequalities.
Adding and Subtracting Rational Expressions
Rational Equations.
2 Equations, Inequalities, and Applications.
A rational expression is a quotient of two polynomials
Solving Equations Containing Rational Expressions § 6.5 Solving Equations Containing Rational Expressions.
Section 6 – Rational Equations
Bell Ringer What is the restriction on rational expressions which results in extraneous solutions? 2. When solving equations with fractions,
Concept 5 Rational expressions.
Solving Rational Equations
Section 4.2 Solving a System of Equations in Two Variables by the Substitution Method.
Presentation transcript:

Lesson 4-6 Rational Equations and Partial Fractions Objective: To solve rational equations and inequalities To decompose a fraction into partial fractions

Rational Equations Rational Equation – has 1 or more rational expressions. Solve by multiplying each side by the LCD

Rational Equations To solve a rational equation: 1. Find the LCM of the denominators. 2. Clear denominators by multiplying both sides of the equation by the LCM. 3. Solve the resulting polynomial equation. 4. Check the solutions.

Find the LCM. Multiply by LCM = (x – 3). Solve for x. Check. Examples: 1. Solve: . LCM = x – 3. Find the LCM. 1 = x + 1 Multiply by LCM = (x – 3). x = 0 Solve for x. (0) Check. Substitute 0. Simplify. True. 2. Solve: . LCM = x(x – 1). Find the LCM. Multiply by LCM. x – 1 = 2x Simplify. x = –1 Solve. Examples: Solve

Check. x = 3 is not a solution since both sides would be undefined. Example: Solve: . x2 – 8x + 15 = (x – 3)(x – 5) Factor. The LCM is (x – 3)(x – 5). Original Equation. x(x – 5) = – 6 Polynomial Equation. Simplify. x2 – 5x + 6 = 0 Factor. (x – 2)(x – 3) = 0 Check. x = 2 is a solution. x = 2 or x = 3 Check. x = 3 is not a solution since both sides would be undefined. Example: Solve

Decomposing a fraction into Partial Fractions. Sometimes we need more tools to help with rational expressions… We will learn to perform a process known as partial fraction decomposition… To find partial fractions for an expression, we need to reverse the process of adding fractions.

To find the partial fractions, we start with The expressions are equal for all values of x so we have an identity. The identity will be important for finding the values of A and B.

To find the partial fractions, we start with Multiply by the LCD So, If we understand the cancelling, we can in future go straight to this line from the 1st line.

This is where the identity is important. The expressions are equal for all values of x, so I can choose to let x = 2. Why should I choose x = 2 ? ANS: x = 2 means the coefficient of B is zero, so B disappears and we can solve for A.

This is where the identity is important. The expressions are equal for all values of x, so I can choose to let x = 2. What value would you substitute next ? ANS: x = - 1 so that the first term becomes 0.

This is where the identity is important. The expressions are equal for all values of x, so I can choose to let x = 2. So,

Example 2 Express the following as 2 partial fractions. Solution: Let Multiply by :

Decomposing Fractions Decompose into partial fractions 2/(x+2) +4/(x-5)

Rational Inequalities To solve rational inequalities: Find the zeros and mark on a number line. Find any exclusions (restrictions) and mark on a number line. Test a value on each interval.

Rational Inequalities Solve Set to 0 LCD=15b Find the zeros

Rational Inequalities -1/15 Test b<-1/15 try b=-1 True Test -1/15<b<0 try b=-1/30 False Test b>0 try b=1 True

Rational Inequalities So

Rational Inequalities Solve -2<x<-1, -1<x<1,1<x<3, x>3