Rational Expressions and Equations Unit 7 Unit Planner

Transcription

1 MAT 100 Armstrong Rational Expressions and Equations Unit 7 Unit Planner 7.1 Simplifying Rational Expressions Read pages p. 547 # 7. Multiplying and Dividing Rational Expressions Read pages p. 556 # 7.3 Adding and Subtracting Rational Expressions with Like Denominators; LCD Read pages Day 1 p. 565 # Day p. 566 # 7.4 Adding and Subtracting with Unlike Denominators Read pages p. 573 # 7.5 Simplifying Complex Fractions Read pages p. 581 # 7.6 Solving Rational Equations Read pages p. 590 # 7.7 Problem Solving Using Rational Equations Read pages p. 596 # 7.8 Proportions and Similar Triangles Read pages p. 608 # 7.9 Variation Read pages p. 618 # Unit 7 Summary/Review Read pages p. 6 # Unit 7 Test [1]

2 NOTES: []

3 Unit 7 Rational Expressions and Equations 7.1 Simplifying Rational Expressions A is an expression of the form B A where A and B are polynomials and B 0. Evaluating Rational Expressions. To evaluate rational expressions we will and follow order of operations. Example. Evaluate x 1 x 1 for x = -3. Finding Values that Make Rational Expressions Undefined. To find the values that make a rational expression undefined we will: 1.. Find all real numbers for which each rational expression is undefined: a. 7x x 5 b. x 1 x x 6 [3]

4 Simplifying Rational Expressions. * To simplify a monomial fraction, we will remove the greatest common factor (GCF) from the numerator and denominator and cancel. * To simplify a polynomial fraction, we will: 1. Simplify:. a. 1x y 14xy b. x 3x 3x 9 c. x 13x 1 x 144 d. 3 x x x 1 e. 5 7 x 3 5 x 3 7 f. x x 3x 1 3 x 3 [4]

5 Simplifying Rational Expressions that have Opposite Factors. * If the terms of two polynomials are the same, except they have opposite signs,: 1.. Simplify, if possible: a. a 1 1 a b. y 1 3 3y c. t t 8 8 [5]

6 Unit 7 Rational Expressions and Equations 7. Multiplying and Dividing Rational Expressions Multiplying Rational Expressions. Steps: Multiply: a. x 3 5 b. 7 5 c. 9 3x 3 y x d. t 1 t 1 t t e. 35x y z 7y z 5xy f. x x x x 4 x g. x 3x x x x x 6 x x [6]

7 4 h. x x 1 3a 1 i. 63 x j. 5 a 7x a x x x 8x 7 k. x 7 Dividing Rational Expressions. Steps: 1.. Divide: a. a b. 9x 15x 35y 14 c. x x x x 1 3x 15 6x 30 x 3x d. 4 x x 1 [7]

8 Converting Units of Measure. A roll of carpeting is 1 feet wide and 150 feet long. Find the number of square yards of carpeting on the roll. The speed with which light moves through space is about 186,000 miles per second. Express this speed in miles per hour. [8]

9 Unit 7 Rational Expressions and Equations 7.3 Adding and Subtracting w/ Like Denominators; LCDs Adding and Subtracting with Like Denominators. Steps: Add: a. x 3x 3x y x b x 5xy c. 3x 1 8x 1 5x 10 5x 10 Subtract: d. 5x x 5x 1 4x e. 3 3 x 3 x 3 f. 5x 1 4x x 3 x 3 g. 3x 1 5x x 1 x x 1 x x 1 x x 1 [9]

10 Finding the Least Common Denominator (LCD). Steps: Find the LCD of each pair of rational expressions. a and 8x 18x b. 0 and x 4x 9 x c. x x and 7x 7 5x 5 d. x 6 x 15x and 8x 16 x 16 [10]

11 Building Rational Expression into Equivalent Expressions. * To build a fraction into an equivalent expression we will always multiply by a form of. Write each rational expression as an equivalent expression with the indicated denominator: a. ; 15 7n denominator 3 6x 30n b. ; x 4 denominator x 4x 4 x 1 c. ; x 6x denominator x x 6x [11]

12 Unit 7 Rational Expressions and Equations 7.4 Adding and Subtracting w/ Unlike Denominators Adding and Subtracting Rational Expressions w/ Unlike Denominators. Steps: Add or subtract. a. 4x 3x 5 11 b b 18b c. x 4 x 5 x 4x d. x 3 x 1 x e. a a 1 a 1 [1]

13 f. a 1 a 4a 4 a 4 4b g. b a 5 * When adding or subtracting rational expressions whose denominators are opposite signs, we will multiply the top and bottom of one fraction by. a. 3 x x y y x b. 3 1 xy x y xy [13]

14 Unit 7 Rational Expressions and Equations 7.5 Simplifying Complex Fractions A rational expression whose numerator and/or denominator contain fractions is called a. Simplifying Complex Fractions Simplify: a. 5x 3 x 9 3 b. 1 1 x x c. 6 y x 6 x y d y 8 y 4y [14]

15 e x 1 [15]

16 Unit 7 Rational Expressions and Equations 7.6 Solving Rational Equations A is an equation that contains one or more rational expressions. Solving Rational Equations. Steps: Solve. a. x b x x c. 3a 1 8 x 1 d. 1 5 a a x 3 x x 3 [16]

17 e. 4 4y 50 y f. 5 5y 5 x x x 1 Solving for a Variable within a Rational Expression Formula. The formula is used in electronics to calculate parallel resistances. Solve it for r. r r1 r [17]

18 Unit 7 Rational Expressions and Equations 7.7 Problem Solving Using Rational Equations Solving Number Problems. If the same number is added to both the numerator and the denominator of the fraction 5 3, the result is 5 4. Find the number. Solving Shared-Work Problems. An inlet pipe can fill an oil tank in 7 days, and a second inlet pipe can fill the same tank in 9 days. If both pipes are used, how long will it take to fill the tank? [18]

19 Solving Uniform Motion Problems. A coach can run 10 miles in the same amount of time as his best student-athlete can run 1 miles. If the student can run 1 mile per hour faster than the coach, how fast can the student run? Solving Investment Problems. At one bank, a sum of money invested for 1 year will earn $96 interest. If invested in bonds, that money would earn $108, because the interest rate paid by the bonds is 1% greater than that paid by the bank. Find the bank s rate. * Hint: Use the formula rt I P where I is the interest, P is the principle (amount invested), r is the annual rate of interest, and t is the time in years. [19]

20 Unit 7 Rational Expressions and Equations 7.8 Proportions and Similar Triangles Writing Ratios and Rates in Simplest Form. Ratios enable us to compare numerical quantities. To prepare fuel for a lawnmower, gasoline must be mixed with oil in the ratio of 50 to 1. To make 14-karat jewelry, gold is mixed with other metals in the ratio of 14 to 10. In the stock market, winning stocks might outnumber losing stocks in the ratio of 7 to 4. A is the quotient of two numbers or the quotient of two quantities that have the same units. Translate each phrase into a ratio written in fractional form: a. The ratio 5 to 9. b. 1 ounces to pounds A is a quotient of two quantities that have different units. A is a mathematical statement that two ratios or two rates are equal. Proportions. Determine whether each equation is a proportion. a b [0]

21 Solve: c. 1 3 a 1 10 d. 18 x 4 8 e. a 4 a f. If 6 apples cost $1.38, how much will 16 apples cost? g. A scale is a ratio (or rate) that compares the size of a model, drawing, or map to the size of an actual object. The scale on a model carousel is 1 inch to that of 160 inches on the actual carousel. How wide should the model be if the actual carousel is 35 feet wide? 1 1 h. A recipe for rhubarb cake calls for 1 cups of sugar for every cups of flour. How many 4 cups of flour are needed if the baker intends to use 3 cups of sugar? [1]

22 Using Proportions to Solve Problems Involving Similar Triangles. Similar triangles have the same. In order to have the same shape, the triangles must have angle pairs with the same measure. If two triangles are similar, all pairs of corresponding sides are in. Example. A tree casts a shadow 18 feet long at the same time as a woman 5 feet tall casts a shadow 1.5 feet long. Find the height of the tree. []

23 Unit 7 Rational Expressions and Equations 7.9 Variation If the value of one quantity depends on the value of another quantity, we can often describe that relationship using the language of variation: The sales tax on an item varies with the price. The intensity of light varies with the distance from its source. The pressure exerted by the water on an object varies with the depth of the object beneath the surface. Solving Direct Variation Problems. * Two variables are said to vary directly if one is a constant multiple of the other. Two variables that vary directly are represented by the equation: where k is a constant (number) called the constant of variation. y kx y From this formula we can simply calculate the constant of variation, k, using k. x Steps: Suppose y varies directly as x. If y = 1 when x = 4, find y when x = 6. [3]

24 The weight of an object on Earth varies directly with its weight on the moon. If a rock weighs 5 pounds on the moon and 30 pounds on Earth, what would be the weight on Earth of a larger rock weighing 6 pounds on the moon? Solving Inverse Variation Problems. * Two variables are said to vary inversely if one is a constant multiple of the reciprocal of the other. Two variables that vary inversely are represented by the equation: where k is a constant (number) called the constant of variation. k y x Suppose y varies inversely as x. If y = 5 when x = 0, find y when x = 50. The volume occupied by a gas varies inversely with the pressure placed on it. That is, the volume decreases as the pressure increases. If a gas occupies a volume of 15 cubic inches when placed under 4 pounds per square inch (psi) of pressure, how much pressure is needed to compress the gas into a volume of 10 cubic inches? [4]