Transcript x = -9
Solving Linear Equations and Inequalities • Solving algebraically • Solving graphically • Solving equations in more than one variable • Solving linear inequalities • Solving double inequalities • Solving absolute value equations • Applications • Solving Algebraically Example: Solve 2x = x – 9 2x – x = x – 9 – x (get x on 1 side) x = -9 (simplify) • Solving Graphically Graph left hand side of equation and right hand side of the equation and see where the graphs meet. y = 2x and y = x -9 x = -9 (just need x-value of point) • Solve 2x + 1 = -x –9 2x + 1 + x = -x –9 + x (x on 1 side) 3x + 1 = -9 (simplify) 3x + 1 – 1 = - 9 – 1 (get x on own) 3x = -10 (simplify) x = -10 3 Check: Replace x with answer. 2(-10/3) + 1 = -(-10/3) – 9 -20/3 + 1 = 10/3 – 9 -17/3 = -17/3 Both sides equal so answer is correct. • Graph both sides x = -3.34 (no y-value required) • Solve 2x = 2x – 3 2x – 2x = 2x – 3 – 2x 0 = -3 (impossible) So no answer! • Graph Lines parallel so no intersection •Solve 2(x + 1)=2x + 2 2x + 2 = 2x + 2 (distribute) 2x + 2 – 2x = 2x + 2 – 2x 2 = 2 (always true) So every x is an answer! Lines coincide, so they intersect everywhere. (same line) Example: Solve algebraically. 7 1 12 3 2x 2 x 7 1 12 2 x( 3 ) multiply by LCD 2x 2 x 7 6 x x 24 simplify 7 6 x 6 x x 24 6 x get x' s on one side 7 7 x 24 simplify 7 24 7 x 24 24 get x on it' s own 17 7 x simplify 17 7 x 7 7 17 x 7 Solve: x = -2.5 7 1 12 3 2x 2 x Words of caution: Solving graphically will give you answers that are imprecise. If you want accuracy you need to solve algebraically. If I ask you to solve an equation I want a precise answer. However, you can see if you are in the ball park by graphing. Solving equations in more than one variable Solve for C 9 F C 32 5 9 F 32 C 32 32 5 9 F 32 C 5 5 59 ( F 32) C 9 95 5( F 32) C 9 5( F 32) C 9 Solving Linear Inequalities A linear inequality is similar to a linear equation except it is an inequality. Here are some examples of linear inequalities. 2x 3 3 5 x 2( x 5) 2 x 3 x 2 x 1 2 x 43 4( x 2) Solve the linear inequality and graph on a number line. 3( x 3) 2 x 5 3x 9 2 x 5 Simplify 3x 9 2 x 2 x 5 2 x Get x' s on 1 side x 9 5 Simplify x 9 9 5 9 Get x on it' s own x 14 Solved -14 (-, -14) in interval notation Solve the inequality and graph the solution on the number line. 3 x 2 11 3 x 2 2 11 2 3x 9 3x 9 divided by a negative flip inequality 3 3 x 3 -3 ( -, -3 ] in interval notation Solve the inequality and graph solution on number line. 1 (9 x 27) 5 3 3x 9 5 3x 14 14 x divided by negative so flip inequality 3 2 x 4 3 4 2 3 [ -4 2/3, ) in interval notation Solving a Double Inequality Solve and graph on a number line. 5 x 5 25 5 5 x 5 5 25 5 trying to get x by itself 10 x 20 so we take 5 away from every part. -10 20 ( -10, 20 ] Solving Absolute Value Equations Solve this absolute value equation. x 5 Almost everyone has a hard time with these equations. The most common error is to only give one solution. When in fact there are usually two answers. Let’s try solving these graphically first. We will graph the left and right hand sides of the equation and see where the graphs meet. y x and y 5 y=|5| y=5 x=-5 x=5 Because the graph of an absolute value function is generally a ‘V’, there is a good chance that you will get two answers. Solving Absolute Value Equations Algebraically Solve the absolute value equation. x 7 Break equation into two pieces The piece x7 or The - piece x7 x 7 2 x 10 12 piece - piece 2 x 10 12 2 x 22 (2 x 10) 12 2 x 10 12 x 11 2x 2 x 1 Solve the following inequality 5t 6 2t 9 piece - piece 5t 6 2t 9 5t 2t 3 3t 3 t 1 (5t 6) 2t 9 5t 6 2t 9 6 7t 9 15 7t 15 t 7 Solving Absolute Inequalities Graphically Solve the absolute inequality graphically. x 1 5 The absolute value function is larger than y = 5 when x is >= 4 and x <= -6 or ( - , -6 ] [ 4, ) -6 4 Solve the absolute inequality graphically 2x 4 x y = | 2x – 4 | x=4 y=x x 4/3 The absolute value function is less than the x function when x is less than 4 and greater than 4/3 4/3 4 or ( 4/3, 4 ) Now let’s solve algebraically x8 5 piece - piece x8 5 x 3 ( x 8) 5 x 8 5 x 13 x 13 -13 -3 So x must be less than –3 and greater than -13 (-13, -3) Solve 2x 5 x piece 2x 5 x x 5 0 x5 - piece (2 x 5) x 2x 5 x 3x 5 0 3 x 5 5 x 3 Answer in interval notation: ( - , -5/3 ] [ 5, ] or 5/3 5 Applications Break Even Analysis I have decided to go into the business of making custom made tile top tables. The fixed cost of setting up my business is $1,000. Each table costs $75 to make. I plan on selling the tables for $115 each. How many tables will I need to sell in order to break even? Solution: There are two functions here. A cost function C(x) and a revenue function R(x) (where x is number of tables sold. C(x) = 1000 + 75x and R(x) = 115x The break even point is the point where cost and revenue are equal. So set the functions equal to each other and solve. 1000 + 75x = 115x 1000 + 75x – 75x = 115x – 75x 1000 = 40x 1000 ÷ 40 = 40x ÷ 40 25 = x So I will have to sell at least 25 tables to break even. More on solving graphically ANSWERS a. x=-5 & 1 & 5 b. (- , -5] [ 1, 5 ] c. (-5,1) (5, ) d. (-6,-2) (2, ) e. [-4,1] [6,) 1. f(x) = g(x) 2. f(x) – g(x) = 0 3. f(x) > 0 4. f(x) > g(x) 5. g(x) < 0 6. f(x) < g(x) Solutions: 1-a 4-c 2-a 5-d 3-e 6-b