B Apply

Add or subtract. Simplify where possible. State any restrictions on the variables.

31. 3 4 x − 2 x 2 fraction 3 , over 4 x end fraction , minus , fraction 2 , over x squared end fraction
32. 3 x + 1 + x x − 1 fraction 3 , over x plus 1 end fraction . plus . fraction x , over x minus 1 end fraction
33. 4 x 2 − 9 + 7 x + 3 fraction 4 , over x squared , minus 9 end fraction . plus . fraction 7 , over x plus 3 end fraction
34. 5 x x 2 − x − 6 − 4 x 2 + 4 x + 4 fraction 5 x , over x squared , minus x minus 6 end fraction . minus . fraction 4 , over x squared , plus 4 x plus 4 end fraction
35. 3 x + x 2 + 5 x x 2 − 2 3 x plus . fraction x squared , plus 5 x , over x squared , minus 2 end fraction
36. 5 y y 2 − 7 y − 4 2 y − 14 + 9 y fraction 5 y , over y squared , minus 7 y end fraction . minus . fraction 4 , over 2 y minus 14 end fraction . plus , 9 over y
37. Think About a Plan For the image of the overhead projector to be in focus, the distance d i d sub i  from the projector lens to the image, the projector lens focal length f, and the distance d o d sub o  from the transparency to the projector lens must satisfy the thin-lens equation 1 f = 1 d i + 1 d o . 1 over f , equals , fraction 1 , over d sub i end fraction , plus , fraction 1 , over d sub o end fraction , .  What is the focal length of the projector lens if the transparency placed 4 in. from the projector lens is in focus on the screen located 8 ft from the projector lens?
    • Can you write the equation for the unknown variable?
    • What units would you use for the focal length of the lens?
38. Optics To read small font, you use a magnifying lens with the focal length 3 in. How far from the magnifying lens should you place the page if you want to hold the lens at 1 foot from your eyes? Use the thin-lens equation from Exercise 37.
39. Open-Ended Write two complex fractions that simplify to x − 2 x + 4 . fraction x minus 2 , over x plus 4 end fraction . .
40. Writing Explain how factoring is used when adding or subtracting rational expressions. Include an example in your explanation.

Simplify each complex fraction.

41. 2 x + 3 y − 5 x + 7 y fraction 2 over x , plus , 3 over y , over negative 5 over x , plus , 7 over y end fraction
42. 1 + 2 x 2 + 3 2 x fraction 1 plus , 2 over x , over 2 plus , fraction 3 , over 2 x end fraction end fraction
43. 1 x y − 1 y 2 1 x 2 y − 1 x y 2 fraction fraction 1 , over x y end fraction , minus , fraction 1 , over y squared end fraction , over fraction 1 , over x squared , y end fraction . minus . fraction 1 , over x , y squared end fraction end fraction
44. 2 x + 4 + 2 1 + 3 x + 4 fraction fraction 2 , over x plus 4 end fraction . plus 2 , over 1 plus . fraction 3 , over x plus 4 end fraction end fraction
45. Harmony The harmonic mean of two numbers a and b equals 2 1 a + 1 b . fraction 2 , over 1 over eh , plus , 1 over b end fraction . .  As you vary the length of a violin or guitar string, its pitch changes. If a full-length string is 1 unit long, then many lengths that are simple fractions produce pitches that harmonize, or sound pleasing together. The harmonic mean relates two lengths that produce harmonious sounds. Find the harmonic mean for each pair of string lengths.
    1. 1 and 1 2 1 half
    2. 3 4 3 fourths  and 1 2 1 half
    3. 3 4 3 fourths  and 3 5 3 fifths
    4. 1 2 1 half  and 1 4 1 fourth

C Challenge

46. Electricity The resistance of a parallel circuit with 3 bulbs is 1 1 R 1 + 1 R 2 + 1 R 3 . fraction 1 , over fraction 1 , over r sub 1 end fraction , plus , fraction 1 , over r sub 2 end fraction , plus , fraction 1 , over r sub 3 end fraction end fraction . .

    1. Find the resistance of a parallel circuit with 3 bulbs that have resistances 5 ohms, 4 ohms, 2.5 ohms.
    2. The resistance of a parallel circuit with 3 bulbs is 1.5 ohms. Find the resistances of the bulbs, if two of them have equal resistances, while the resistance of the 3rd is 3 ohms less.
47. Show that the sum of the reciprocals of three different positive integers is greater than 6 times the reciprocal of their product.

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