DIVIDING BY ZERO. Rational Expressions and Equations. Note Package. Name: 1: Simplifying Rational Expressions 2: Multiplying and Dividing

Transcription

1 MAT30S Mr. Morris Rational Expressions and Equations Lesson 1: Simplifying Rational Expressions 2: Multiplying and Dividing 3: Adding and Subtracting 4: Solving Rational Equations Note Package Extra Practice Page 527 #4-6, 7ac, 8ab, 10a, 11a, 12 Page 537 #3ab, 6-8, 11, 12ab Page 553 #3cd, 5a, 6a, 7bd, 8bd, 9b, 10, 11d Page 566 #3ab, 5bd, 6b, 7, 8, 10bc, 11, 12a Page 584 #5-10 5: Word Problems Page 597 #3-8 Name: DIVIDING BY ZERO

2 Page 1 of 33 Lesson 1 Objective: Recall about rational numbers: - A fraction when the numerator and denominator are integer values (excluding 0 for the denominator) 3 4 and 2 5 are rational numbers In Grade 11 math we extend rational numbers to include rational expressions Definition: Rational Expressions - Same basic definition as a rational number - Uses polynomials in the numerator and denominator - Denominator still cannot be equal to 0 3x 2 4x 3, x 2 4 x 2 x 12, and 4x x 2 +2 are all rational expressions

3 Page 2 of 33 - No variables in radicals or exponents in these expressions! Evaluating rational expressions is easy, substitute the values for the variables inside the rational expressions and calculate the value Example 1: Evaluate the rational for x = 2 and y = 2 2x 3y 2 3x 2y Since we know that rational numbers cannot have 0 as their denominator we must talk about non-permissible values, or NPVs

4 Page 3 of 33 Definition: Non-Permissible Values - Values that variables cannot equal because it will make a denominator equal to 0 - Can be equal to a number, or equal to an relationship with another variable. x+5 x 3 has an NPV of x = 3 since that would make the denominator 0 - To find NPVs we set the denominator to equal 0 then solve for the value that satisfies the equal - All of the ways we have used in the past to factor will still be present here - Difference of square, trinomial factoring, etc

5 Page 4 of 33 Example 2: Find the NPVs for the following rational expressions a) 5x x 2 9 b) 3x+2 x 2 8x+16

6 Page 5 of 33 A large part of this unit will be simplifying rational expressions. - Try to factor the top and bottom (whichever method) and state NPVs. - Do not state NPVs after simplifying! Always before! - Common multiplying terms must be present in both numerator and denominator to simplify a rational expression 20xy 8x 2 6xy 18y 2y

7 Page 6 of 33 Example 3: Write each rational expression in simplest form. Whiteboard work a) 25a3 b 2 c 35ab 5 b) 6x2 +12x 3x

8 Page 7 of 33 c) x 2 49 x 2 5x 14 d) 2x2 5x 3 9 x 2

9 Page 8 of 33 End lesson 1 Practice sheets Page 527 #4-6, 7ac, 8ab, 10a, 11a, 12

10 Page 9 of 33 Lesson 2 Objective: Just like we can multiply rational numbers together we will learn to multiply rational expressions together. Recall the rules of multiplying fractions together Multiply the two numerators together and the two denominators together For our rational expressions there is a couple more steps involved 1) Factor all expressions (if possible) 2) State all NPVs (both rational expressions) 3) Multiply numerator, multiply denominators but no not distribute bracketed terms 4) Divide out any like factors (simplify) 5) State final answer, with restrictions

11 Page 10 of 33 x x + 10 x 1 x2 1 x + 9

12 Page 11 of 33 Example 1: Simplify each expression and state restrictions a) 3b2 5a 2a 9 b) 2x2 (x+2) 3x 5(x 4) 8x(x+2)

13 Page 12 of 33 c) x2 x 20 x 2 6x x2 12x+36 x 2 +9x+20

14 Page 13 of 33 Recall dividing rational numbers uses a specific technique that flips the second fraction and changes the division into a multiplication The next step is to look at dividing rational expressions. Again, the same rules apply as dividing rational numbers, but with a few more steps involved. - Factor and state restrictions of all denominators - Take the reciprocal of the second rational (flip it) - State the restrictions on the new denominator of the second rational (new denominator) - Multiply numerators, multiply denominators - Divide out any like factors - State answer, with all restrictions

15 x + 1 x x + 10 x2 1 x + 9 Page 14 of 33

16 Page 15 of 33 Whiteboard/desk work Example 2: Simplify and state the restrictions a) 5(x 3) 2x 10(x 3) 3x(x+5) b) 4 x 2 3x 15 5x 10 x 5

17 Page 16 of 33 c) 6x 2x 2 +3x 9 8x4 4x 2 9

18 Page 17 of 33 End lesson 2 Practice sheets Page 537 #3ab, 6-8, 11, 12ab

19 Page 18 of 33 Lesson 3 Objective: Adding and subtraction rational numbers always relies on finding a common denominator - The lowest common denominator is preferred but a fool proof method is to just multiple the 2 denominators together to obtain a common denominator Recall the method to add and subtract rational numbers

20 Page 19 of 33 - Simplifying always happens after the addition or subtraction When making the transition to rational expressions we add a couple of steps to the process - Determine the common denominator between both fractions (factoring may occur for this to happen) - State NPVs - Write both fractions over the common denominator by multiplying numerator and denominator appropriately - Add or subtract normally - Simplify as needed

21 Page 20 of ab 3 4b 2 p 1 p 2 + p + 3 p + 1

22 Page 21 of 33 Simplify the following Rational Expressions* a) 3c 2 3cd c+8 5c 2 d b) a a

23 Page 22 of 33 c) x 2 49 x 2 +14x+49 d) n 3 n 2 +3n 18 n 2 n 2 +n 20

24 Page 23 of 33 End lesson 3 Practice sheets Page 553 #3cd, 5a, 6a, 7bd, 8bd, 9b, Page 566 #3ab, 5bd, 6b, 7, 8, 10bc, 10, 11d

25 Page 24 of 33 Objective: Lesson 4 Recall solving linear equation and quadratic equations: Starting Example: Solve the equations x = 7 x2 4x = 5

26 Page 25 of 33 Solving rational equations uses all the techniques we ve discussed with expressions, except now you are actually solving for a value 1) Factor and state restrictions (NPVs) 2) Try to knock out denominators by cross multiplying or multiplying every term by a common denominator 3) Solve for variable 4) Verify that solution is not an NPV x x + 2 = 5 7

27 Page 26 of 33 Example 1: Solve each of the following equations a) x+1 = x 1 x 1 x+3 b) 1 2x 2 5 = 1 10x

28 Page 27 of 33 Example 2: Solve each of the following a) 10 x 2 12x+35 = x x 5 b) x+1 x+6 + x 2 x+4 = 11x+32 x 2 +10x+24

29 Page 28 of 33 End lesson 4 Practice sheets Page 584 #5-10

30 Page 29 of 33 Objective: Lesson 5 Using rational equations in word problems usually boils down to distance and time type of problems but others can exist. - List the information given - Form an equation - List the NPVs - Solve and verify the solutions A good working knowledge of using ratios is needed for these questions. You will have to build them on your own and there is more than one way to do it.

31 Page 30 of 33 Example 1: A boat travels at an average speed of 15 km hr in still water. The boat travels 12km downstream in the same time as it travels 8km upstream. Determine the average speed of the current.

32 Page 31 of 33 Example 2: Marissa can paint a garage door in 3 hours. When Marissa and Roger work together, they can paint the same garage door in 1 hour. How long would it take roger to paint the garage door on his own?

33 Page 32 of 33 Example 2: On a canoe trip, Patan paddled upstream a distance of 10km. On the return trip downstream, the average speed of the canoe was 5 km greater than its h speed upstream. Write, then simplify an expression for Patan s total paddling time in terms of the average speed upstream.

34 Page 33 of 33 End lesson 5 Practice sheets Page 597 #3-8