411 3.7 Variation 17. x4 = 35 - 2x2 18. 16 x - 14 x2 = 2 19. x + 2 = 3x + 1 20. 5x + 4 + 2 = 9 21. 6x - 7 63 + 4x 22. 5x + 3 79 - 8x 23. 1 - 2x Ú 15 24. 7 … 11 + 3x 25. -11 64x + 5 69 26. 5 Ú 3 - 2x Ú 2 27. 3x + 5 Ú 8 28. 4x - 3 67 29. 8 - 2x 73x2 30. 2x2 + 15 7 -11x 31. -21x + 60 … 9x2 32. 2x2 - 63 … -11x 33. x - 1 1x - 322 = 0 34. x x2 - 4 70 35. 2x4 - 3x3 + 20 740x2 - 21x 36. 6x4 + 27x - 18 … 25x3 - 14x2 37. 25x2 - 20x + 4 70 38. 3x + 4 … x2 39. x4 - 2x3 - 3x2 + 4x + 4 = 0 40. x3 + 5x2 + 3x - 9 Ú 0 41. -x4 - x3 + 12x2 = 0 42. -4x4 + 13x2 - 3 70 43. 2x2 - 13x + 15 x2 - 3x Ú 0 44. x2 + 3x - 1 x + 1 73 The direct variation equation y = kx defines a linear function, where the constant of variation k is the slope of the line. For k 70, • As the value of x increases, the value of y increases. • As the value of x decreases, the value of y decreases. When used to describe a direct variation relationship, the phrase “directly proportional” is sometimes abbreviated to just “proportional.” Direct Variation To apply mathematics we often need to express relationships between quantities. For example, • In chemistry, the ideal gas law describes how temperature, pressure, and volume are related. • In physics, various formulas in optics describe the relationship between the focal length of a lens and the size of an image. When one quantity is a constant multiple of another quantity, the two quantities are said to vary directly. For example, if an hourly wage is $10, then 3pay4 = 10 # 3hours worked4. Doubling the hours doubles the pay. Tripling the hours triples the pay, and so on. This is stated more precisely as follows. 3.7 Variation ■ Direct Variation ■ Inverse Variation ■ Combined and Joint Variation Direct Variation y varies directly as x, or y is directly proportional to x, if for all x there exists a nonzero real number k, called the constant of variation, such that y =kx.
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