Winter,

CH4-1 Inequalities and Their Graphs Background: Many times we don’t know the answer but we certainly know what range we need or want. For example, nurses want to see body temperatures of what? Nurses might look body temperatures to be LESS than or equal to 98.6 °F. Speed limits allow us to drive LESS than 70 mph but GREATER than 45 mph. 2

CH4-1 Inequalities and Their Graphs Vocabulary for SYMBOLS: < means…. LESS THAN (mouth closed to smaller quantity) > means….. GREATER THAN (mouth opens to bigger quantity) ≤ means…. LESS THAN OR EQUAL TO (mouth closed to smaller qty) ≥ means…. GREATER THAN OR EQUAL TO (mouth opens to bigger qty) ① means… The number 1 is NOT INCLUDED ❶ means…. The number 1 IS INCLUDED 3

CH4-1 Inequalities and Their Graphs How To Use It: Ex.1 Determine whether each number is a solution of the given inequality. -1 > xa. 0b. -3c. -6 a. -1>0 Is this true? NO! b. -1>-3 Is this true? YES! c. -1 > -6 YES! 4

CH4-1 Inequalities and Their Graphs When in doubt, put it on the number line and doublecheck! 5

CH4-1 Inequalities and Their Graphs Now, you do ODDS,

4-2 & 4-3 Solving Inequalities So, how do you solve inequalities? Same as you did with = sign in CH3!! ALWAYS FOLLOW YOUR RECIPE!!!! Recipe to Solve Equations Step1: Get x term(s) alone on one side = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 7

4-2 & 4-3 Solving Inequalities How To Use It: Ex.1 Solve each inequality. Check your solution. n – 7 ≥ n ≥ 9 9 – 7≥ 2 2 ≥2 Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 8

4-2 & 4-3 Solving Inequalities How To Use It: Ex.2 Solve each inequality. Check your solution. a ≤ a ≤ a ≤ a ≤ -4 Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 9

CH4-2 & 4-3 Solving Inequalities Now, you do: 4-2: Evens 2-20, 22, : ODDS,

CH4-4 Solving Multi-Step Inequalities What if there are variables on both sides of the inequality sign? What do we do then? Same as CH3! Use the recipe to solve for the variable. Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 11

CH4-4 Solving Multi-Step Inequalities How To Use It: Ex.1 Solve each inequality. 2(3+3g) ≥ 2g + 14 PEMDAS starts it off… 6 + 6g ≥ 2g g ≥ -2g 6 + 4g ≥ ≥ -6 +4g ≥ g ≥ 2 Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 12

CH4-4 Solving Multi-Step Inequalities How To Use It: Ex.2 Write and solve an inequality that models each situation. Suppose it costs $5 to enter a carnival. Each ride costs $1.25. You have $15 to spend at the carnival. What is the greatest number of rides that you can do? First, define variable(s): r= number of rides $5 = entry fee (to be added to cost of rides) $15 = total cost Next, start writing sentences as math equation Total cost = entry fee + cost of rides 13

CH4-4 Solving Multi-Step Inequalities How To Use It: Ex.2 Write and solve an inequality that models each situation. Suppose it costs $5 to enter a carnival. Each ride costs $1.25. You have $15 to spend at the carnival. What is the greatest number of rides that you can do? Next, plug-in what you know into this equation. Total cost = entry fee + cost of rides $15 = $5 + $1.25∙r But now, look at the = sign is that right? No, we know the MAX we can spend is $15 so the right side of that equation better be LESS THAN or EQUAL TO THAT 14

CH4-4 Solving Multi-Step Inequalities How To Use It: Suppose it costs $5 to enter a carnival. Each ride costs $1.25. You have $15 to spend at the carnival. What is the greatest number of rides that you can do? So, what sign do we use? ≥ $15 ≥ $5 + $1.25∙r -5 ≥ ≥ 1.25∙r 1.25 ≥ ≥ r You can buy NO MORE THAN 8 RIDES 15

CH4-4 Solving Multi-Step Inequalities Now, you do: Evens

CH4-5 Compound Inequalities Background: Sometimes, we want a range for the answer, not just one value. What do we do when this happens? How do we solve something like: -4 < t+2 < 4 Nothing is different than before! You still want to isolate your variable, using your recipe…. 17

CH4-5 Compound Inequalities How To Use It: Ex.1 Solve each inequality. -4 < t+2 < 4 Steps 1-3 are done -4 < t+2 < < t < 2 Graph it on a number line to see if this result makes sense Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 18

CH4-5 Compound Inequalities Now, you do: Odds

CH4-6 Absolute Value Equations and Inequalities Background: When you have absolute value bars, you have two possible solutions, a positive and a negative. Ex.1 |x| = 6 x can be +6 x can also be -6 So you have to switch the = sign for an inequality and make the number negative, to get answers. So it is easiest to just write two equations and solve for the two answers. 20

CH4-5 Compound Inequalities How To Use It: Ex.2 Solve each inequality. |3c-6| ≥ 3 First, to get rid of Absolute Value bars, Rewrite as two equations. 3c -6 ≥ 33c-6 ≤ -3 Now solve each equation and combine into one answer Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 21

CH4-5 Compound Inequalities How To Use It: 3c -6 ≥ 33c-6 ≤ c ≥ 9 3c ≤ c ≥ 3 c ≤ +1 c ≤ +1 or c ≥ 3 Recipe to Solve Equations Step1: Get x term(s) alone on one side of = sign. Step2: Combine Like Terms. Step3: Isolate x using opposite functions. Step4: Plug x value back in to original question and check answer. 22