MATH 150 Pre-Calculus Fall, 2014, WEEK 3 JoungDong Kim Week 3: 2B, 3A Chapter 2B. Solving Inequalities a < b a is less than b a b a is less than or equal to b a > b a is greater than b a b a is greater than or equal to b Notation 1. (a,b) represents the set of all real numbers x such that a < x < b. (All numbers between a and b) 2. [a,b] represents the set of all real numbers x such that a x b. (including two endpoints of interval) 3. The set (, ) is used to represent all real numbers. In summary open parentheses mean exclude the endpoint and square brackets mean include the endpoint. Ex1) Which of the following numbers ( 7, 5, 3, 1,0) are in the interval [ 5, 1), 1
Linear Inequalities To solve a linear inequality, isolate the variable in the same fashion as solving a linear equation. However, there is one major difference: Whenever you multiply or divide an inequality by a negative number, you must switch the inequality symbol (or direction). Rules for Inequalities (A,B and C: real numbers) 1. A B A+C B +C 2. A B A C B C 3. If C > 0, then A B CA CB 4. If C < 0, then A B CA CB 5. If A > 0, and B > 0 then A B 1 A 1 B 6. If A B and C D, then A+C B +D 2
Ex2) Solve the inequality 3x < 9x+4 Ex3) Solve for y: y 1 3 3y 5 > 1 3
Pair of Simultaneous Inequalities a < x < b means the number x is between a and b. More formally, we say that a < x and x < b. Ex4) Solve the inequalities 4 3x 2 < 13. 4
Ex5) Solve for x: x+4 < 2x 7 < 3 3x. 5
Absolute Value Inequalities We use the following properties to solve inequalities that involve absolute value. Properties of Absolute Value Inequalities 1. x < c c < x < c 2. x c c x c 3. x > c x < c or x > c 4. x c x c or x c Ex6) Solve the inequality x 5 < 2 6
Ex7) Solve the inequality 3x+2 6 2 Ex8) Solve the inequality 3x 4 +6 3 7
Nonlinear Inequalities If an inequality is not linear, 1. Move all terms to one side. If necessary, rewrite the inequality so that all nonzero terms appear on one side of the inequality sign. If the nonzero side of the inequality involves quotients, bring them to a common denominator. 2. Factor. Factor the nonzero side of the inequality. 3. Find the intervals. Determine the values for which each factor is zero. These numbers will divide the real line into intervals. List the intervals determined by these numbers. 4. Solve. To determine which, test a number in the interval by substituting it into the inequality. If the chosen number satisfies the inequality, then so does every other number in the interval, and if the number does not satisfy the inequality, then no other number from that interval will satisfy the inequality. Ex9) Solver the inequality x 2 5x+6 0 8
Ex10) Solve for x: x 2 4x > 12 Ex11) Solve for x: x 4 +3x 3 4x 2 0 9
Ex12) Solve: 1+x 1 x 1 Ex13) Solve for x: x < 2 x 1 10
Ex14) Solve: x+4 x 14 < 2 x 6 11
Chapter 3. Graphing Chapter 3A. Rectangular Coordinate System Rectangular Coordinate System (or Cartesian Coordinate System) (x,y) is called the coordinates of the point, which gives the horizontal and vertical distance of the point from the origin respectively. The point of intersection of the x-axis and the y-axis is the origin, (0,0), and the two axes divide the plane into four Quadrants, labeled I, II, III, and IV. Ex15) Plot the following points on a rectangular coordinate system. A(2, 3), B(0, 5), C( 4, 1), D(3, 0), E( 2, 4) 12
Ex16) Shade the regions given by each set a) { (x,y) x 0 } b) { (x,y) x > 1 } c) { (x,y) x > 1, 1 y 2 } 13
Definition Any set of ordered pairs is called a relation. The plot of every point associated with an ordered pair in the relation is called the graph of the relation. The set of all first elements in the ordered pairs is called the domain of the relation. The set of al second elements in the ordered pairs is called the range of the relation. In Ex15) In Ex16-a) In Ex16-c) 14
Distance Formula The distance between the points A(x 1,y 1 ) and B(x 2,y 2 ) in the plane is d(a,b) = (x 2 x 1 ) 2 +(y 2 y 1 ) 2 15
Ex17) Determine whether the points A( 2,1), B(5,1) and C(0,3) are the vertices of a right triangle 16
Midpoint Formula The midpoint of the line segment that connects points A(x 1,y 1 ) and B(x 2,y 2 ) is the point M(x,y) with coordinates ( x2 +x 1, y ) 2 +y 1 2 2 Ex18) Find the midpoint of the line segment that connects the points ( 2,9) and (7, 3). 17
Ex19) If ( 5,8) is the midpoint of the line segment connecting A(3, 2) and B, find the coordinates of the other endpoint B. 18
Circles The set of all points P(x,y) that are a given fixed distance from a given fixed point is called a circle. The fixed distance, r, is called the radius and the fixed point C(h,k) is called thecenter. Equation of a circle with center (h,k) and the radius r is (Standard form) (x h) 2 +(y k) 2 = r 2 If center is (0,0) x 2 +y 2 = r 2 19
Ex20) Find the center and radius, and graph each equation a) x 2 +y 2 = 25 b) (x 2) 2 +(y +1) 2 = 25 Ex21) Write an equation for the circle with C(3, 5) and radius 2. 20
General form of an Equation of a Circle x 2 +y 2 +cx+dy +e = 0 Note. To find the center and radius of a circle that is in general form, we must write the equation in standard form. (by Completing the Square) 21
Ex22) Find the center and radius of the circle x 2 + y 2 + 2x 6y + 6 = 0. Graph the circle. Find domain and range. Ex23) Find the center and radius of the circle 3x 2 +3y 2 6x+12y+2 = 0. 22
Ex24) If A( 7,2) and B(5,8) are coordinates of a diameter of a circle. Write the equation for the circle. 23