3.1/3.2: Solving Inequalities and Their Graphs

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1 3.1/3.2: Solving Inequalities and Their Graphs - Same as solving equations - adding or subtracting a number will NOT change the inequality symbol. - Always go back and check your solution - Solutions are a set of numbers, meaning that there are answers that work Symbols Meaning Buzz Words > < Examples: Solve and Check the following 1) x ) x 5 < 8 3) x (-4) > 6 Graphing Steps: 1) make sure inequality is solved for. (get variable on the left side) WHY? 2) Plot the value that you solved for on a number line and label the number line 3) Open dot for greater than or less than / Closed dot for the or equal to 4) Shade in the appropriate direction Examples: Graph the solutions from above. 4) x ) x 5 < 8 6) x (-4) > 6

2 Examples: Write inequalities for the following, then solve. 7) Four times a number is no more than three times that number plus eight. 8) A number decreased by eight is at least 14. Story Problem: An entrepreneur decides to open up a yogurt shop in Avon. He decides to sell his yogurt by the pound. He charges $.75 for every pound, p that he sells. He spends $150 on supplies every day. If he wants to make a profit, how much does he need to sell each day? Story Problem: The area of a rectangle is at least 54in 2. If the length is 6 times the width, find each measurement. Write an inequality that represents this situation and solve using square roots. Length = Width =

3 3.3: Solving By Multiplication and Division - Same as solving equations - The inequality sign FLIPS when multiplying or dividing by a NEGATIVE - Graphing solutions is done in the same way Examples: Solve the following inequalities and graph. 1) 5 x 6 2) 5x 35 3) 3 x < 2 4) 2y > 5 5) -3x < 15 6) x 5-2 7) x 9 2 8) x > 8

4 3.4: Solving Multi-Step Inequalities - same as equations, move by addition/subtraction first, then multiply or divide - rules for multiplying/dividing by negatives still apply - treat the inequality like an equals sign until you multiply or divide a negative ALL REAL Solutions---- NO SOLUTION ---- Examples: Solve and graph the following inequalities 1) 2y - 5 < 7 2) 5 - x > 4 3) 3 (x + 2) < 7 4) 4x - 3 2x ) 8(x + 2) 3(x 4) < 5(x 7) + 8 Story Problem: Jarrod leaves every morning with $13 in his pocket. He travels to work on 465 and gets to work at 8:15 every morning. If Jarrod wants to have at least $150 in his pocket at the end of the day, how many hours, h, should he work if he gets paid $12 per hour. Write an inequality and solve it.

5 3.6: Solving Compound Inequalities - Two conditions (either and or or ) And Statements: -Looking for where two graph intersect, this set of numbers must satisfy Inequalities If I ask, Do you like pizza and brussel sprouts? ---You must like both to say yes. Or Statements -Looking for the set/sets that satisfy either inequality, DOESN T HAVE TO BE BOTH If I ask, Do you like pizza or brussel sprouts? ---You only need to like one to say yes. Note the bottom line shows the overlap. That is what you want on an AND statement. Note the bottom line shows everything above. That is what you want on an OR statement. Examples: Solve the following and graph (Hint: When no word exists and there are two inequality symbols, re-write it using the word and with 2 inequalities) 1) x 4 > -5 and x 4 < 2 2) 4 < 2x + 6 < 10 3) x + 4 < 12 or x 3 > 18

6 4) 2x > 6 or x 5 < 12 5) 9-3x ) -5 < x 4 < 2 7) To earn a B in Algebra, you must average between an 84 and 86% on your tests. You scored 86, 85, and 80 on your first 3 tests. What possible scores can you earn on the last test to earn a B in the course? What do you need to know how to do? What do you already know? Where are you trying to get to?