Math 141. Chapter 1: Equations and Inequalities. Notes. Name...

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1 Math 141 Chapter 1: Equations and Inequalities Notes Name... 1

2 1.1 The Coordinate Plane Graphing Regions in the Coordinate Plane 1. Example: Sketch the region given by the set (a) {(x, y) x = 3} (b) {(x, y) y = 4} 2

3 (c) {(x, y) 2 < x 2} (d) {(x, y) x > 3 and y 3} 3

4 Distance Formula 2. Example: Consider the points (5, 0) and (0, 4). (a) Plot the points in the coordinate plane. (b) Find the distance between the points. 4

5 Midpoint Formula 3. Example: Find the midpoint of the segment that joins (5, 0) and (0, 8). 4. Example: If M(3,4) is the midpoint of the line segment AB and if A has coordinates (-1,2), find the coordinates of B. 5

6 Shifting the Coordinate Plane 5. Example: Suppose that each point in the coordinate plane is shifted 4 units to the right and 3 units downward. (a) The point (3,2) is shifted to what new point? (b) The point (a, b) is shifted to what new point? Reflecting in the Coordinate Plane 6. Example: Suppose that the y-axis acts as a mirror that reflects each point to the right of it into a point to the left of it. (a) The point (1,3) is reflected to what point? (b) The point (a, b) is reflected to what point? 6

7 1.2 Graphs and Equations in Two Variables Graphing Equations by Plotting Points 1. Example: Sketch the graph of 3x y = 5. Plot 5 points. 2. Example: Sketch the graph of y = x 2 3. Plot 5 points. 7

8 3. Example: Sketch the graph of y = x. Plot 5 points. Intercepts 4. Example: Find the x and y intercepts of y = x Example: Find the x and y intercepts of x y2 25 = 1 Graph the equation. 8

9 6. Example: Find the intercepts of the equation whose graph is shown

10 Circles Standard form of an equation of the circle with center (h, k) and radius r: Standard form of an equation of the circle with center (0, 0) and radius r: 7. Example: Graph x 2 + y 2 = 9 8. Example: Graph (x 2) 2 + (y + 1) 2 = 4 10

11 11

12 9. Example: Find an equation of the circle with radius 10 and center (1,-3) 10. Example: Show that the equation x 2 + y 2 2x + 6y 15 = 0 represents a circle. Find the center and radius of the circle. Symmetry Symmetry with respect to the y-axis Test: The equation must be unchanged when x is replaced with x Graph: Symmetry with respect to the x-axis Test: The equation must be unchanged when y is replaced with y Graph: Symmetry with respect to the origin Test: The equation must be unchanged when (x, y) is replaced with ( x, y) Graph: 12

13 11. Example: Test y = x 5 9x 3 for symmetry. 12. Example: Test y = x 2 + 2x for symmetry. 13

14 Slope of a line 1.3 Lines Point-Slope Form Slope-Intercept Form 1. Example: Find the slope of each line. (a) The line passes through (1,2) and (5,8) (b) y = 3 2 x + 4 (c) y = 4 (d) x = 5 14

15 2. Example: Find an equation of the line through (-3,1) and (-4,3). Parallel Lines Two nonvertical lines are parallel if and only if they have the same. 3. Example: Find an equation of the line through (2,5) that is parallel to the line 6x + 4y + 5 = 0 Perpendicular Lines Two lines with defined slopes are perpendicular if and only if their slopes are. A vertical line is to a horizontal line. 4. Example: Find the equation of the line that is perpendicular to the line 6x+4y+5 = 0 and passes through (1,1). 15

16 1.5 Modeling with Equations Investement Problems Interest=Principle Interest Rate Time (in years) I = P rt 1. Example: Melissa won $60,000 on a slot machine in Las Vegas. She invested part at 2% simple interest and the rest at 3%. In one year she earned a total of $1,600 in interest. How much was invested at each rate? Solution: Define a variable. Let x= Organize relevant data into a table or a chart 1st account 2nd account Together principal rate = interest Equation: 16

17 Mixture Problems 2. Example: A chemist needs 180 ml of a 55% solution but has only 22% and 76% solutions available. Find how many ml of each should be mixed to get the desired solution. Solve this problem by using only one variable. Solution: Define a variable. Let x= Organize relevant data into a table. Solution 1 (high %) Solution 2 (low %) Mixture (middle %) Percent ml = Amount of Solution Equation: Solve! 17

18 Work Problems The standard equation that will be needed for these problems is Portion of job done in given time=work rate time spent working When two or more objects are working, the setup becomes or work rate of object 1 + work rate of object 2 =one job done time spent working time spent working time together time apart time together + time apart =one job done 3. Example: An office has two envelope stuffing machines. Machine A can stuff a batch of envelopes in 5 hours, while Machine B can stuff a batch of envelopes in 3 hours. How long would it take the two machines working together to stuff a batch of envelopes? Solution: Define a variable. Let t time together time apart Equation: time together + time apart =one job done 18

19 4. Example: Mary can clean an office complex in 5 hours. Working together John and Mary can clean the office complex in 3.5 hours. How long would it take John to clean the office complex by himself? Solution: Define a variable. Let t Motion Problems The standard formula that we ll be using here is Distance=Rate Time 5. Example: Two cars are 500 miles apart and moving directly towards each other. One car is moving at a speed of 100 mph and the other is moving at 70 mph. Assuming that the cars start moving at the same time how long does it take for the two cars to meet? Solution: Draw a diagram. Define a variable. Let t Organize relevant data into a table. 19

20 Car 1 Car 2 Total Equation: Solve! Rate Time Distance 20

21 1.6 Quadratic Equations Solving using factoring The zero factor property: If ab = 0 then either a = 0 and/or b = 0 1. Example: Solve by factoring (a) 4m 2 1 = 0 (b) 10z z + 6 = 0 (c) 5x 2 = 2x 21

22 Completing the Square ( ) 2 b If you start with x 2 + bx and add, you ll get a factorable trinomial: 2 ( ) 2 ( b x 2 + bx + factors as x + b ) This process is called completing the square. 2. Example: Use completing the square to solve each equation. (a) x 2 + 6x + 1 = 0 (b) 2x 2 + 7x + 2 = 0 22

23 Quadratic Formula If ax 2 + bx + c = 0, then x = b ± b 2 4ac. 2a 3. Example: Solve 1 3 x2 + 5 x = 1 using the quadratic formula. 3 Applications of Quadratic Equations 4. Example: An object is thrown or fired straight upward at an initial speed of v 0 ft/s will reach a height of h feet after t seconds, where h and t are related by the formula h(t) = 16t 2 + v 0 t. Suppose that a bullet is shot straight upward with an initial speed of 800 ft/s. (a) When does the bullet fall back to ground level? (b) When does the bullet reach a height of 6,400 ft? (c) When does it reach a height of 1 miles? (d) How high is the highest point the bullet reaches? 23

24 1.7 Solving Other Types of Equations Polynomial Equations 1. Example: Solve x 3 = 16x 2. Example: Solve 5x 3 5x 2 10x = 0 3. Example: Solve 2x 3 + x 2 18x 9 = 0 24

25 Rational equations 4. Example: Solve. 10 x 12 x 3 = 4 5. Example: Solve. x 2x + 7 x + 1 x + 3 = 1 25

26 Radical Equations 6. Example: Solve each equation. (a) 4 6x = 2x (b) x x = 8 26

27 Equations quadratic in form 7. Example: Solve each equation. (a) x 4 5x = 0 Equations with Fractional Powers 8. Example: Solve each equation. (a) x 4 3 = 4 (b) x 4 3 5x = 0 27

28 1.8 Solving Inequalities To solve an inequality, we must isolate the variable on one side of the inequality symbol. When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol reverses. 1. Example: Solve each inequality. Give the solution set in set-builder notation and interval notation. (a) 3t + 11 < 1 (b) 1 4 z 1 2 2z (c) 2 < 3x

29 Quadratic Inequalities 2. Example: Solve 2x 2 x > 1 3. Example: Solve (x 3) 2 (x + 3) 3 < 0 Rational Inequalities 4. Example: Solve x + 1 x Example: Solve 3 + x 3 x > 1 29

30 1.9 Absolute Value Equations and Inequalities Equations Rule: expression = a expression=a or expression= a where a 0 Before you can use the rule above, you must isolate the absolute value on the left side. 1. Example: Solve each equation. (a) x + 2 = 4 (b) 3 2x = 8 (c) x + 2 = 2x 1 30

31 Absolute value inequalities Rule: x > c x > c OR x < c This rule applies to x c as well. Rule: x < c c < x < c This rule applies to x c as well. 2. Example: Solve each inequality. Give the solution set in interval notation. (a) x + 1 > 4 (b) 2x 3 4 (c) x (d) x + 3 < 2 (e) 2x

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1) Recitation (1.1) Question 1: Find a point on the y-axis that is equidistant from the points (5, 5) and (1, 1) Question 2: Find the distance between the points P(2 x, 7 x) and Q( 2 x, 4 x) where x 0. Question