9. x2 5 14. x > 11.8 -2 20. 3x- 9 < 6 22. 5 - x 2 -4 24. -(1 – x) > 2x – 1 26. -7x + 3<4 - x 21.x + 4 10 23.- x> x 25. 2 +x 2 3(x – 1) MOORBBA 28. -3 < -x< 2 27-< x< 4 30. 3 < x + 4 < 10 32. 100 + x< 41 – 6x < 121 + x 3. 29. -7 2x² – 18 43 (x + 1)(x – 2)(x – 4) < 0 45. (x² – 1)(x² – 4) < 0 40. (3x + 2)(x – 1) < 0 42. 4x > 9x + 9 44. (1 – x)(x +)(x – 3) <0 46. (x – 1)²(x + 3)(x – 5) = 47 5 x + 8 10 48. х — 3 50. 49. N-1 51. + 2 > 0 - 52. x + 3 x(x - 1) 53. (1 + x)(1 – x) 54. - x +5 х CHAPTER 1 INEQUALITIES, EQUATIONS, AND GRAPHS 19. The points (-2, 0). (-2, 6), and (3, 0) are vertices of a rectangle. Find the 20. Describe the set of all points (x, r) in the coordinate plane. The set of all In Problems 21-26, sketch the set of points (x, y) in the xy-plane whose coordin ISIGNIA fourth vertex. points (x, -x). Des Moines 22. ху > 0 24. x 2 and y 2-1 26. y 1 satisfy the given conditions. %3D 23. x s1 and yl s2 25. x| > 4 Kansas City St. Lo In Problems 27–32, find the distance between the given points. 28. A(-1, 3), B(5, 0) 30. A(-12, -3), B(-5, –7) 32. A(-3, 4), B(-3, -1) 27. A(1, 2), B(-3, 4) 29. A(2, 4), B(-4,-4) 31. A(-}, 1), B(}, -2) FIGURE 1.3.15 Map f In Problems 33-38, determine whether the points A, B, and C are vertices of atigh 'A triangle, an isosceles triangle, or both. 34. A(-2, -1), B(8, 2), C(1, -11) 36. A(4, 0), B(1, 1), C(2, 3) 38. A(1, 1), B(4, 5), C(8, 8). 33. A(8, 1), B(-3, –1), C(10, 5) 35. A(2, 8), B(0, –3), C(6, 5) 37. A(-2, 1), B(0, 9), C(3, 4) 39. Suppose the points A(0, 0) and B(0, 6) are vertices of a triangle. Find a third FIGURE 1.3.1 Problem 63 УА vertex C so that the triangle is equilateral. 40. Find all points on the y-axis that are 5 units from the point (4, 4). 41. Consider the line segment joining the points A(-1, 2) and B(3, 4). (a) Find an equation that expresses the fact that a point P(x, y) is equidistan S-2MA speg FIGURE from A and from B. Probler (b) Describe geometrically the set of points described by the equation in nam. 42. Use the distance formula to determine whether the points A(-1, -5), B(2,4, and C(4, 10) lie on a straight line. 010.-10.0.043. Find all points each with x-coordinate 6 such that the distance from each poim (-1,2) is V85. 44. Which point, (1/V2, 1/V2) or (0.25, 0.97), is closer to the origin? (- In Problems 45 and 46, find all points P(x, x) that are the indicated distance fm the given point. (o 45. (-2, 0); V10 46. (3, –5); V34 In Problems 47–52, find the midpoint M of the line segment joining the poins A and B. 1. 47. A(4, 1), B(-2, 4) 49. A(-1,0), B(-8, 5) 51. A(2a, 3b), B(4a, –6b) 48. A(}, 1), B(3. –3) 50. A(}, –}), B(-3, 1) 52. A(x, x), B(-x, x + 2) In Problems 53-56, find the point B if M is the midpoint of the line segment joinm points A and R.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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