NAME DATE PERIOD Solve Rational Equations A rational equation contains one or more rational epressions. To solve a rational equation, first multipl each side b the least common denominator of all of the denominators. Be sure to eclude an solution that would produce a denominator of zero. Eample 0 + Solve 0 + + =. Check our solution. + = Original equation 0( + ) ( 0 + = 0( + ) + ) ( ( + ) + (0) = 4( + ) Multipl. + + 0 = 4 + 4 Distribute. = - = - Divide each side b. Check Eercises 0 + + = 0 + - + 8 0-0 0 = ) Multipl each side b 0( + ). Subtract 4 and from each side. Original equation = - Simplif. Solve each equation. Check our solution.. - + 6 Stud Guide and Intervention =. 4t - - 4 - t 4. m + m + m - m = 4. 4 - = + =. + 6. - - = 4 - + 4 - = 0 7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa can travel miles upstream or 6 miles downstream in the same amount of time. What is the speed of her motorboat in still water? Is this a reasonable answer? Eplain. 8. WORK Adam, Bethan, and Carlos own a painting compan. To paint a particular house alone, Adam estimates that it would take him 4 das, Bethan estimates das, and Carlos 6 das. If these estimates are accurate, how long should it take the three of them to paint the house if the work together? Is this a reasonable answer? Lesson Chapter 8 Glencoe Algebra
Solve Rational Inequalities To solve a rational inequalit, complete the following steps. Step Step Step Stud Guide and Intervention (continued) State the ecluded values. Solve the related equation. Use the values from steps and to divide the number line into regions. Test a value in each region to see which regions satisf the original inequalit. Eample Solve n + 4 n. Step The value of 0 is ecluded since this value would result in a denominator of 0. Step Solve the related equation. n + 4 n = Related equation n ( n + n) 4 = n( ) Multipl each side b n. 0 + = 0n Simplif. = 0n Add.. = n Divide each side b 0. Step Draw a number with vertical lines at the ecluded value and the solution to the equation. - - - 0. Test n = -. Test n =. Test n =. ) - + (- 4 is true. + 4 is not true. + 4 is true. The solution is n < 0 or n.. Eercises Solve each inequalit. Check our solutions.. a + 4. - > 4. 4.. p + 4 p > 4 - + < 6. - + > - Chapter 8 40 Glencoe Algebra
Solve each equation. Check our solution... - = = -6 Skills Practice. = 4 n + 4. - z = z Lesson. d + = d - 6. r - = 8 r 7. + + = 8. - = - 7. + - 7 + = 0. b - b + = 4 - b + b -. = q + q q +. 8-4 z = 8z - 8 z +.. 7. n + + n - = n - - 8 + + + = - + f f - 4 + f - = f + 4. 6. 8. Solve each inequalit. Check our solutions.. - + 4 > + + 0. - v < v w + + w - = 4 w - 4 p + p + 7p + - p + = p + 4 8 t - + 4 t + = t - 0. k - 4 k > 0. n + n < n. m - m < - 4. < - Chapter 8 4 Glencoe Algebra
Solve each equation or inequalit. Check our solutions.. + 4 =. p + 0 p - = 4 p. - = 7. t < t +. 4 w - = - w + Practice. - - = s 4. s + + s = s + 8 s + - - 6. - + = 0 8. h + h = h - 0. - a < 7 a. 4 + 0 <. 4 p + p <. g + g g - = g - 7. n + + n - = n - 4. k - + 4 k - 4 = k - 7k +. + + 7 - = 4 - - 0 r. r + 4 + 4 r - 4 = r + 6 r - 6. 8 + > 6 4. - = 4 - + + b 6. b + b - = - b - b - 8. c + c - = 4 - c - c - 4v 0. v - - v v - = v - v +. + 4-4 + - = + 4. = 6a - a + 7 + a + 7. BASKETBALL Kiana has made of free throws so far this season. Her goal is to make 60% of her free throws. If Kiana makes her net free throws in a row, the function f() = + represents Kiana s new ratio of free throws made. How man + successful free throws in a row will raise Kiana s percent made to 60%? Is this a reasonable answer? Eplain. 8. OPTICS The lens equation p + q = relates the distance p of an object from a f lens, the distance q of the image of the object from the lens, and the focal length f of the lens. What is the distance of an object from a lens if the image of the object is centimeters from the lens and the focal length of the lens is 4 centimeters? Is this a reasonable answer? Eplain. Chapter 8 4 Glencoe Algebra
Word Problem Practice. HEIGHT Serena can be described as being 8 inches shorter than her sister Malia, or as being.% shorter than Malia. In other words, 8 H + 8 = 8, where H is Serena s height in inches. How tall is Serena?. CRANES For a wedding, Paula wants to fold 000 origami cranes. 4. PROJECTILES A projectile target is launched into the air. A rocket interceptor is fired at the target. The ratio of the altitude of the rocket to the altitude of the projectile t seconds after the launch of the rocket is given b the t formula. At what time - t + 40t + 7 are the target and interceptor at the same altitude? Lesson She does not want to make anone fold more than cranes. In other words, if N is the number of people enlisted to fold cranes, Paula wants 000 N. What is the minimum number of people that will satisf this inequalit?. RENTAL Carlos and his friends rent a car. The split the $00 rental fee evenl. Carlos, together with just two of his friends, decide to rent a portable DVD plaer as well, and split the $0 rental fee for the DVD plaer evenl among themselves. Carlos ends up spending $0 for these rentals. Write an equation involving N, the number of friends Carlos has, using this information. Solve the equation for N.. FLIGHT TIME The distance between John F. Kenned International Airport and Los Angeles International Airport is about 00 miles. Let S be the airspeed of a jet. The wind speed is 00 miles per hour. Because of the wind, it takes longer to fl one wa than the other. a. Write an equation for S if it takes hours and minutes longer to fl between New York and Los Angeles against the wind versus fling with the wind. b. Solve the equation ou wrote in part a for S. c. Write an equation and find how much longer it would take to fl against the wind between New York and Los Angeles if the wind speed increases to 0 miles per hour and the airspeed of the jet is miles per hour. Chapter 8 4 Glencoe Algebra
Enrichment Asmptotes in Three-Dimensions In the same wa that lines can be asmptotes to a graph in two-dimensions, a plane can be an asmptote to a three-dimensional graph. The curve graphed b z = + has no asmptotes. The value of z approaches negative infinit as and approach negative infinit; the value of z approaches positive infinit as and approach positive infinit. On the other hand, the curve z = + does have an asmptote and hole. The curve is undefined for the planes = 0 and for = 0. The plane z = 0 is an asmptote because the curve approaches this plane as and grow infinitel positive or as and grow infinitel negative. z Eample z = +. Identif points of discontinuit and asmptotes for the curve Because and are the denominators of fractions, the three-dimensional function must be undefined where undefined for the planes = 0 and for = 0. In addition, as and grow infinitel positive or as and grow infinitel negative, z approaches but does not reach 0. Thus the plane z = 0 is an asmptote. Eercises Identif points of discontinuit and asmptotes for the following curves.. z =. z =. z = + + + + + 8 + + + 4 + 4. z = + Chapter 8 44 Glencoe Algebra