Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = 4 0 0 4 = 0 0 Definitions Rational Epression: A fraction that contains variables. In mathematical epressions such as 6 or 5 6 you can use any number for. Mathematically speaking, the epression is defined for all values of. In other words, there are no restrictions on what we can use for. Rational epressions, on the other hand, can have restrictions on what we can plug in for the variable(s). There are certain values of the variables that will make the epression undefined which means it does not have a definite value. When is a rational epression undefined and why? State the restrictions for each rational epression. 5.. 7 5. y 4. 9 5. 7 0 ~ ~
Simplifying a rational epression means cancelling or dividing out any common factors in the numerator and denominator. We ADD like terms. In rational epressions we cancel common factors NOT like terms. Rules for simplifying rational epressions Monomials (One term) ) Since monomials contain multiplication only, and multiplication is the same operation as the division indicated by the fraction bar, you can cancel any common factors. 6 y y 4 Polynomials (More than one term) ) Since polynomials contain addition/subtraction, they are not the same operation as division. Therefore, you must factor polynomials before you can find the common factors to cancel. Parenthesis around polynomials can help you remember this. 6 6 State the restrictions on each rational epression and then simplify.. 5 5 6. 6 4 6. 7 4. 5y 0 4y y y ~ ~
5. 4y 6y 5 6. r 5r 0 4r Continue simplifying these rational epressions. You do not have to find the restricted values. 7. 9 5 4 8. 6 4 9. ac ab bc b a ab b 0. 9 y 9 6y y ~ ~
Lesson #0 Absolute Value Equations A.A. Solve absolute value equations and inequalities involving linear epressions in one variable Absolute value: a number s distance from zero on a number line. If you know that the absolute value of a number is 0 or in other words that the number is 0 units away from zero on the number line, that number could be or. If 0, what could equal? We solve more comple equations in the same way. Eample) 7.. If necessary isolate the absolute value. (SADMEP). Get rid of the absolute value sign using its inverse: set the epression in the absolute value equal to the answer and the (answer).. Solve each resulting equation and write your solution in roster notation. 4. Check your solutions in the original equation. A common mistake is to change the sign in the absolute value. Do not do this. Always change the sign of the answer. If you know that the absolute value of a number is - or in other words that the number is - units away from zero on the number line, that number could be or. Eample : 4 ~ 4 ~
Practice. Solve each equation.. 7. 4 4 8. 4 4. 5 9 5. 6 8 6. 9 0 7. 4 5 8. 6 ~ 5 ~
Lesson # Multiplying (and Dividing) Rat. Ep. A.A.6 Multiplying Fractions: Perform arithmetic operations with rational epressions and rename to lowest terms Simplify the following problem without cross canceling: 9 5 5 8 = Simplify the following problem by cross canceling first: 0 4 7 = Imagine doing the problem, 5 4 0 5, without cross cancelling. You would have to use the distributive property in both the numerator & denominator. It would take forever. Q: Why can you cross cancel when multiplying rational epressions? A: Rational epressions are division. For eample, 5 0 is the same as 5 0. Since multiplication and division are really the same operation, you can cancel or divide out common factors before you multiply. Steps for Multiplying Rational Epressions 5 4 ) Factor each numerator and denominator 0 5 (State the restricted values if asked to) ) Cancel any factors that appear in one of the numerators and one of the denominators. ) Multiply the remaining terms in the numerator. Multiply the remaining terms in the denominator. 4) Check to see if you can reduce any further. E) 4 ~ 6 ~
Division Dividing is the same as. For eample, 5 is the same 5. This eample helps you to remember to,, when dividing with fractions or rational epressions. Note: With division, the restricted values will come from the two original denominators as well as the new denominator on the second fraction after changing the problem to multiplication. ) 4 9 9 ) 8 y y 6y 6y 9y ) 6 8 8 9 8 9 4) a 5 ( a 5) a 0 4a 0 5) y y 4y y ~ 7 ~
Lesson # Adding and Subtracting Rat. Ep. A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Adding is really combining things that are the same or : For each problem, simplify either term first if possible. Decide if the terms are like terms. Write no if they are not. If they are like terms write the name of the terms and combine them together. 5 5 5 5 5 y 5 5 7 y 5y 48 7 5 48 What are like terms with fractions? 5 = 4 5 = The denominator: The numerator: ~ 8 ~
Day : Monomial Denominators Least Common Monomial Denominators: a. 4, 7 4 b. 5, 6 8 c., 4 6 d., y 4 5 0 e., 4 f. y, 7 y, 6 4y g.,, 4 y y z z h., 4 Steps for Adding and Subtracting Rational Epressions 0. State the restricted values.. Find the LCD and write it on the side. 6 y. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) Multiplying the numerator and denominator by the same value is the same as multiplying the fraction by. Therefore its value has not changed. 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! ~ 9 ~
) 4 = ) 6 = ) b a a b = y y 4) y = 5) Which epression is equivalent to () a b () a b a b () a b (4) 6) The sum of, 0 5, is () () 5 5 5 () 5 (4) 5 5 a b? This is not meant to confuse you. The problem is just telling you what the restricted value is. 7) What is the sum of 7n and 7 n? () n () 0 n () 4 n (4) 58 n ~ 0 ~
Day : Binomial Denominators Nothing changes ecept now you have to think about binomial factors. Remember your. The factors of 5 are: The factors of 5 are: The factors of 0 y are: The factors of 0 are: Remember, when finding the LCD, you are simply making sure every factor in both denominators is present. Least Common Binomial Denominators: a. 4, b. 5, c., 6 6 d., 4 e. 4, f. y 7 y 6,, 7 4 y y y y g., 4 6 h. 4, ~ ~
Steps for Adding & Subtracting Rational Epressions Nothing has changed! 0. State the restricted values.. Find the LCD and write it on the side. 5. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! ) 8 ) = 9 ~ ~
) ( ) = 4) 8 4 4 = 5) 6 6 = 4 6) = 4 5 7) 9 6 6 ~ ~
Lesson # Rational Equations A.A. Solve rational equations and inequalities Good news: With rational equations, we can always get rid of the fractions, turning them into regular quadratic or linear equations. Denominators in rational epressions are really what you are dividing by. To get LCD rid of division in an equation, multiply both sides by the. Cross multiplying is basically the same thing because you are multiplying by both denominators. ) 7 ) 7 ) 4 5 4 7 4) 4 When solving a rational equation, any solution is valid UNLESS it is a restricted value. Find the restrictions for each of the following equations.. 7. 7. 4 5 4 7 4. 4 ~ 4 ~
Steps for Solving Rational Equations 4 5. State any restrictions for the rational epressions in the equations. These values cannot be solutions to your equations.. Simplify either side of the equation if possible.. Find the LCD of the denominators on both sides of the equation. 4. Multiply both sides of the equation by the common denominator so that each denominator will cancel. 5. Solve the resulting equation. 6. Check your solutions to make sure that they are not restricted values. Solve each of the following equations. Epress your solutions in roster notation. 6 A. 5 0 B. C. 7 4 49 7 ~ 5 ~
D. 4 4 E. 5 a a a 6 F. a a a b b G. b 5 b 5 ~ 6 ~
Lesson #4 Absolute Value Inequalities A.A. Solve absolute value equations and inequalities involving linear epressions in one variable Review: Steps for Solving Quadratic Inequalities Make the inequality an equation and solve it. 5 6 Set up a number line with the two solutions on it. These are called the critical points. Use open or closed circles depending on the type of inequality. Choose numbers as test points: one smaller than the critical points, one between the critical points, and one larger than the critical points. = = = Check to see if each test point works in the original inequality. Shade the regions on the number line where the test point made the inequality true. Write your solution in either interval notation or set builder notation. When solving absolute value inequalities, most of the steps are the same as quadratic inequalities. These steps that are the same are bolded. Solve the absolute value inequality as if it were an equation. Steps for Solving Absolute Value Inequalities 7 ~ 7 ~
Set up a number line with the two solutions on it. These are called the critical points. Use open or closed circles depending on the type of inequality. = = = Choose numbers as test points: one smaller than the critical points, one between the critical points, and one larger than the critical points. Check to see if each test point works in the original inequality. Shade the regions on the number line where the test point made the inequality true. Write your solution in either interval notation or set builder notation. A) Find and graph the solution set: 5 B) Find and graph the solution set: 4 ~ 8 ~
C) *Find and graph the solution set: 8 D) *Find and graph the solution set: 7 7 5 Thinking questions:. *Which of the following inequalities has (the null or empty set) as its solution set? a. b. c. d.. *Which of the following inequalities has all real numbers as its solution set? a. b. c. d. ~ 9 ~
Lesson #5 Simplifying Comple Fractions A.A.7 Simplify comple fractional epressions Simplify the following rational epressions: 0 = = 4 5 Comple fractions - within. For eample, 6 8 is a comple fraction. Comple fractions are not in simplest form because there are still operations to perform. How to simplify comple fractions Epress in simplest form: a. Perform the addition/subtraction in the numerator if the comple fraction. b. Perform the addition/subtraction in the denominator if the comple fraction. (This can be done at the same time as step a). 6 8 = c. Once there is only one fraction in the numerator and one fraction in the denominator, perform the remaining division: keep, change, flip, and multiply. d. Reduce the resulting fraction if possible. ) 4 ~ 0 ~
) ) z z 5 5 4) r m r m 5) 6) 7 ~ ~
Lesson #6 Solving Rational Inequalities A.A. Solve rational equations and inequalities Solving rational inequalities will be similar to solving quadratic and absolute value inequalities. The main difference is that the critical values will include the solutions to an equation AND the values where the rational epression(s) are undefined. Find the value(s) where the rational epression(s) are undefined. Any undefined values will be critical points. 4 5 4 Solve the rational inequality as if it were an equation. Set up a number line with any solutions or restricted values on it. These are called the critical points. Use open or closed circles depending on the type of inequality. = = = Choose numbers in each interval created the critical points. (With rational inequalities the number of critical points can vary.) Test those numbers in the original inequality. Shade the regions on the number line where the value works in the inequality. Write your solution in either interval notation or set builder notation. ~ ~
. Solve each inequality and epress your solution in either set builder or interval notation. 4 5. 8 4 0. 5 4. 4 ~ ~