Chapter 4 RATIONAL EXPRESSIONS AND EQUATIONS

Chapter – EQUIVALENT RATIONAL EXPRESSIONS

RATIONAL EXPRESSIONS A rational expression is an algebraic fraction with a numerator and a denominator that are polynomials. What is a rational number? What might a rational expression be? Examples:

NON-PERMISSIBLE VALUES What value can x not have? For all rational expressions with variables in the denominator, we need to define the non-permissible values. These are the values for a variable that makes an expression undefined. In a rational expression, this is a value that results in a denominator of zero.

EXAMPLE a)Write a rational number that is equivalent to b)Write a rational expression that is equivalent to a) What are some other equivalent fractions? b) Firstly, what are the non- permissible values? Can I possibly factor either the numerator or the denominator? We need to write our non-permissible values at the end.

EXAMPLE State the restrictions for each rational expression. a)b) a)Determine the non-permissible values: 4 – x = 0 x = 4 There is only one non-permissible value, x = 4. b) Determine the non-permissible values: x 2 – 5x = 0 x(x – 5) = 0  x = 0  x – 5 = 0 x = 5 Two non-permissibles: x = 0, and x = 5 We need to factor.

FACTORING REVIEW When it’s possible to factor a rational expression, we need to be able to do so. Removing a common factorFactoring trinomials If there is a variable or a number that is a common factor in all the terms of an expression, we can “factor it out.” Example: 18x x + 42 We can factor out a 6.  18x x + 42 = 6(3x 2 + 6x + 7) Example: 4x 3 + 6x 2 We can factor out a 2 and an x 2.  2x 2 (2x + 3) x 2 + 9x + 18 What numbers multiply to 18, and add to 9?  3 and 6! x 2 + 9x + 18 = (x + 6)(x + 3) Example: x 2 + 4x – 21 x 2 + 4x – 21 = (x + 7)(x – 3)

EXAMPLE For each of the following, determine if the rational expressions are equivalent. a) b) a)What would I need to multiply 9 by to get –18? The expressions are equivalent! b) Another method is to check using substitution. So, choose a value that you’d like to put in for x. What is a good value to choose?  Let’s try x = 3. The expressions are not equal for x = 3, so they aren’t equivalent.

Independent Practice PG , #3, 5, 6, 11, 14, 15, 16

Chapter – SIMPLIFYING RATIONAL EXPRESSIONS

EXAMPLE Simplify the following rational expression: What are both 24 and 18 divisible by? 66 What are the non- permissible values?

EXAMPLE Simplify the following rational expression: Can I factor the numerator? Can 5x be divided out of 15x 3 ? What are the non-permissible values? Don’t forget your non- permissible values at the end.

EXAMPLE Simplify the following rational expression: Non-permissible values: 3m 3 – 4m 2 = 0  m 2 (3m – 4) = 0 m 2 = 03m – 4 = 0 m = 0 3m = 4 m = 4/3 So, m ≠ 0, 4/3

Independent Practice PG , #3, 4, 5, 8, 9, 10, 13.

Chapter – MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS

EXAMPLE Simplify the following product: Step 1: The first step in rational expression problems is always to factor. Where can we factor here? Step 2: Find the non-permissible values. Look at all of the denominators. Step 3: Multiply the numerators and the denominators together. Note: the non- permissible values stay the same.

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EXAMPLE Simplify each quotient. a)b) Step 1: Factor Step 2: Find the non-permissibles. In division problems, use each denominator and the second numerator. Step 3: Take the reciprocal of the second expression, and then multiply.

EXAMPLE Simplify each quotient. a)b) Step 1: Factor Step 2: Find the non-permissibles. Why didn’t I use 6w, w + 6, or 9w 2 ? Step 3: Reciprocal, and then multiply.

EXAMPLE Simplify the following expression:

Independent Practice PG , #1, 3, 5, 6, 7, 9.

Chapter – ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

EXAMPLE Simplify the following sum: What are the non-permissible values? Find a common denominator. Our common denominator will be 8x 2. What will we need to multiply 4x by to get 8x 2 ?

EXAMPLE Simplify the following difference: Find the non-permissible values: What is the common denominator for this expression?

EXAMPLE Simplify the following expression: Step 1: Factor wherever possible. Step 2: Non-permissible values. Step 3: Find the common denominator. Multiply all the different factors together. Can I factor?

Independent practice PG , #4, 5, 6, 7, 8, 13

EXAMPLE A jet flew along a straight path from Calgary to Vancouver, and back again, on Monday. It made the same trip on Friday. On Monday, there was no wind. On Friday, there was a constant wind blowing from Vancouver to Calgary at 80 km/h. While travelling in still air, the jet travels at a constant speed.Determine which trip took less time. What is the equation for time, when you have speed and distance? How fast is the jet’s airspeed for two different trips on Friday?

EXAMPLE CONTINUED Total for Monday Total for Friday Recall that when you add fractions you need to have a common denominator.

EXAMPLE CONTINUED To be able to compare these two times, we need to have either the numerator or the denominator be the same. Can we easily create a common numerator or denominator? Which denominator is larger?  So what does that say about T 1 ? A larger denominator means that the fraction is smaller. So that means that T 1 is a smaller number, and was a shorter trip.

Independent practice P , #9, 10, 11.

Chapter – SOLVING RATIONAL EQUATIONS

EXAMPLE Solve the following equation for x: You can tell that it is an equation problem and not an expression problem because of the equal sign. There are a different set of rules, so it’s important to differentiate. Step 1: Factor. Step 2: Non-permissibles. Step 3: What would the LCD be? Step 4: Multiply each numerator by the whole LCD. Step 5: Simplify and Solve. You should be able to get rid of the denominators. However, 3 is a NPV. What does this mean?

EXAMPLE When they work together, Stuart and Lucy can deliver flyers to all the homes in their neighbourhood in 42 minutes. When Lucy works alone, she can deliver the flyers in 13 minutes less time than Stuart when he works alone. When Stuart works alone, how long does he take to deliver the flyers? Let y be the time it takes Stuart alone.  then how long does it take for Lucy?  Lucy takes (y – 13) Always consider the fraction of deliveries that can be made in 1 minute: Stuart alone: Lucy alone: Together: Does y = 6 make sense?

EXAMPLE Rima bought a case of concert T-shirts for $450. She kept two T-shirts for herself and sold the rest for $560, making a profit of $10 on each T-shirt. How many T-shirts were in the case? What is the expression for price per t-shirt? What is the expression for profit per shirt? Do both these answers make sense?

EXAMPLE Solve the equation. What are some non-permissible values? The non-permissible values are 2 and -2.

EXAMPLE Solve the equation. What are some non-permissible values? Does k = –2 work? Why or why not?

Independent practice PG , #1, 6, 8, 10, 11, 12, 15