P.1 Solving Equations

Transcription

1 P.1 Solving Equations Definition P.1. A linear equation in one variable is an equation that can be written in the standard form ax+b = 0 where a and b are real numbers and a 0. The most fundamental type of algebra is solving a linear equation: Example P.1.1. Solve ax+b = 0 Lets try something a little harder Example P.1.2. Solve x 5 x 2 = 3+ 3x 10 Step 1: To clear the denominator multiply by the common denominator on both sides. Step 2: Solve for x. ( x 10 5 x 2) = ( 3+ 3x ) x 5x = 30+3x 6x = 30 x = -5 Example P x+1 8x 2x 1 = 4 1

2 Example P x 2 x+3 = 3(x+5) x 2 +3x Quadratic Equations There are several ways to solve quadratic equations: Method 1: The Zero Factor Property If a b = 0 then a = 0 OR b = 0. Example P.1.5. x 2 +5x+6 = 0 Method 2: Square Root Principle Example P.1.6. x 2 32 = 0 If x 2 = c then x = ± c. Example P.1.7. (x 12) 2 = 16 2

3 Method 3: Completing the Square (a+b) 2 = a 2 +2ab+b 2 AND (a b) 2 = a 2 2ab+b 2 so to solve the equation we complete the square: x 2 +bx = c x 2 +bx+ ( ) b 2 = c+ 2 ( ) b 2 2 (x+ b 2 )2 = c+ b2 4 x = b 2 ± c+ b2 4 Example P.1.8. x 2 +4x 32 = 0 Example P x 2 12x = 14 3

4 Method 4: Quadratic Formula To solve the equation we can use the formula: ax 2 +bx+c = 0 Example P x 2 x 1 = 0 x = b± b 2 4ac 2a Example P x = 4 x 2 Polynomials of higher degree Still need to factor but we have to work a little harder. Example P x 3 125x = 0 Example P x = 0 4

5 Example P x 3 +2x 2 3x 6 = 0 For this example we want to factor by grouping. We will group and factor x 3 +2x 2 = x 2 (x+2) and 3x 6 = 3(x+2). Example P x 4 +5x 2 36 = 0 This 4th order polynomial is really a quadratic if we we let u = x 2. Then the equation looks much simpler: u 2 +5u 36 = 0 and we can solve it with our usual techniques. Make sure to solve for x in the end. Equations Involving Radicals Example P x 3 = 0 Step 1: Radical MUST be by itself on one side of the equation. Step 2: Square both sides step 3: Always check to make sure your answers work in the ORIGINAL equation. Example P x+ 31 9x = 5 5

6 Example P y +1 = 5 y Example P x+1 5 = 0 This time cube both sides of the equation. Example P x+5 x 5 = 0 6

7 Example P x 5 3 = x 6 Absolute Value An equation with an absolute value is always TWO equations: x = 4 = x = 4 OR x = 4 Example P x = x 2 +x 3 We start by writing it as two equations: x = x 2 +x 3 x = x 2 +x 3 ALWAYS check your solutions: 7

8 P.2 Solving Inequalities I. Graphing inequalities x > 2 x 2 x 2 AND x 4 x 2 OR x 4 x 2 AND x 4 II. Interval Notation Inequality notation Interval notation x > 2 (2, ) x 2 [2, ) x 2 AND x 4 2 x 4 [2, 4] x 2 OR x 4 (,2] [4, ) 8

9 III. Properties of inequalities (1) Transitive: a < b and b < c = a < c (2) Addition of Constants: If a < b then a+c < b+c (3) Addition: If a < b and c < d then a+c < b+d (4) Multiplication by a constant: If c > 0 and a < b then ac < bc If c < 0 and a < b then ac > bc NOTE: If you multiply or divide by a negative number you reverse the order of the inequality. IV. Solving linear inequalities Example P x < 40 Example P x < 40 Example P (x+1) 2x+3 Example P (3x+5) <

10 V. Absolute value and inequalities Absolute value is still two equations Example P.2.5. x 2 > x 2 > 5 OR x 2 > 5 Example P.2.6. x 7 < 5 5 < x 7 < Question: What does x 2 < 5 mean? Answer: All real numbers within five units of two. So all real numbers within 5 units of 8 would be written as: And all real numbers at least 5 units from 8 would be written as: VI. Solving polynomial inequalities Example P.2.7. x 2 < 5 Step 1: Set equation equal to zero and find the zeros. (Factor) Step 2: Set up a table of signs 10

11 Step 3: Find where the table gives negative values and write the solution Example P.2.8. x 2 +2x 3 0 Example P.2.9. (x 1) 2 (x+2)

12 Example P x3 12x 2 x+2 0 VII. Domain The Domain of a function is the set of x-values where the function is defined. Example P Find the domain of f(x) = x

13 P.3 Graphical Representations of Data Cartesian Coordinates All points in the plane are ordered pairs (x,y) where the 1 st coordinate is directed distance on the x - axis and the 2 nd coordinate is directed distance on the y - axis. The xy-plane is divided into fours quadrants labeled I, II, III, and IV Example P.3.1. At various times, the amount of water in a tub was measured and recorded in the table of values. Sketch a plot of the data. Time Water in tub (min) (gallons) The distance Formula (x 1,y 1 ) (x 2,y 2 ) The distance between two points is given by d = (x 2 x 1 ) 2 +(y 2 y 1 ) 2 13

14 The Midpoint Formula (x 1,y 1 ) (x 2,y 2 ) The midpoint between two points is given by ( x2 +x 1 M.P. =, y ) 2 +y Example P.3.2. Find the distance and midpoint between (1, 12) and (6, 0) Shifting Points (Translating) If you add a quantity to a point it is shifted up and right. If you subtract it is moved left and down. If you add or subtract to just one of the coordinates then you will only get motion in one direction. Example P.3.3. Use the graph below to (1) Move the given triangle right six units. (2) Move the given triangle down 3 units

15 P.4 Graphs of Equations We reviewed in section P.3 how to graph points so now we want to know how to graph equations. Suppose we want to graph the equation y = 2x + 5. This is a relationship between x and y where the value of y is determined by they choice of x. For each x we can find a y value and that is one point (x,y) on the graph: x y = 2x+5-1 (-2)(-1)+5 = 7 0 (-2)(0)+5 = 5 1 (-2)(1)+5 = 3 2 (-2)(2)+5 = 1 5/2 (-2)(5/2)+5 = x and y Intercepts x-intercept: The point where the graph crosses the x-axis. To find the x-intercept you set y = 0. y-intercept: The point where the graph crosses the y-axis. To find the y-intercept you set x = 0. Example P.4.1. Find all intercepts for. y = 4x 3 16x 15

16 Symmetry Symmetric about y-axis A graph is symmetric about the y-axis if it is the same on both sides of the y-axis. Thus when (a,b) is on the graph then ( a,b) is also on the graph. f(x) = f( x) for all x. Symmetric about x-axis A graph is symmetric about the x-axis if it is the same on both sides of the x-axis. Thus when (a,b) is on the graph then (a, b) is also on the graph. Symmetric about the origin A graph is symmetric about the origin if the graph is unchanged by a 180 degree rotation about the origin. Thus when (a,b) is on the graph then ( a, b) is also on the graph. The short version Symmetry The equation is equivalent when... y-axis x is replaced with x x-axis y is replace with y origin x and y are replaced by x and y. 16

17 Example P.4.2. Find the symmetry of y = x 3. Try replace x with x y = x 3 y = ( x) 3 Try replace y with y y = x 3 y = (x) 3 Try replace x with x and y with y y = x 3 y = ( x) 3 Draw a sketch: Since we have origin symmetry we can just plot a few positive numbers. Circles A circle is the set of all points in the plane equidistant from a fixed point. The distance is called the radius and the fixed point is called the center. (h,k) (x,y) Any point on the circle, (x,y), is a distance r from the center (h,k). r 2 = (x h) 2 +(y k) 2 is the standard form of the circle. 17

18 Example P.4.3. Write the equation of the circle with center (-4, 2) and radius r = 5. Example P.4.4. Graph y = x 3 x y = x = = = = = 1 Example P.4.5. Graph y = x 3 Find intercepts first: y - int: set x = 0 x - int: set y = 0 x y = x

19 P.5 Linear Equations in Two Variables The simplest mathematical model is the linear equation in two variables. The standard form is (slope-intercept) y = mx+b where m is the slope and b is the y-intercept. You will recall that slope = rise run. The slope is the amount of vertical change relative to the horizontal change. Sometimes we think of it as the change in y over change in x. To calculate the slope between two points (x 1,y 1 ) and (x 2,y 2 ) the formula is: m = y 2 y 1 x 2 x 1 = y x = rise run. Positive Slope Negative Slope y y x x Example P.5.1. Sketch the graphs of the following two functions: y = 2x+3 y = 1 2 x+3. 19

20 Example P.5.2. Find the slope between the following pairs of points. (a). (-3, 0) and (4, 4) (b). (-3, 1) and (4, 1) (c). (-3, 1) and (-3,4) Point-Slope Form y y 1 = m(x x 1 ) You always need two things: 1. a point: (x 1,y 1 ) AND 2. a slope m. Example P.5.3. Write the equation of the line through (-3,0) and (4, -4). Write the equation in the point slope form and the slope-intecept form. 20

21 Example P.5.4. Write the equation of the lines through (a). (-3, 1) and (4, 1) (b). (-3, 1) and (-3,4) Parallel and Perpendicular Lines Parallel lines have the same slope. If y = m 1 x+b 1 is parallel to y = m 2 x+b 2 then m 1 = m 2. Perpendicular lines have negative reciprocal slopes. If y = m 1 x + b 1 is perpendicular to y = m 2 x+b 2 then m 1 = 1 m 2. Example P.5.5. Write the equations of the lines parallel and perpendicular to 4x + 2y = 3 passing through the point (2, 1). 21