CH4-1 Inequalities and Their Graphs

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1 Fall, Mrs. Kummer Background: Many times we don t know the answer but we certainly know what rangewe 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 5 mph. 2 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) 1means The number 1 is NOT INCLUDED ❶means. The number 1 IS INCLUDED 3 Ex.1 Determine whether each number is a solution of the given inequality. -1 > x a. 0 b. -3 c. -6 a. -1>0 Is this true? Let s check it on the number line NO! b. -1>-3 Is this true? Let s check it on the number line YES! c. -1 > -6 YES! 1

2 When in doubt, put it on the number line and doublecheck! Now, you do ODDS, & -3 Solving Inequalities So, how do you solve inequalities? Same as you did with = sign in CH3!! ALWAYS FOLLOW YOUR RECIPE!!!! Step: Plug x value back in to original question and check answer. -2 & -3 Solving Inequalities Ex.1 Solve each inequality. Check your solution. n n Step3: Get x alone using opposite functions. Step: Plug x value back in to original question and check answer

3 -2 & -3 Solving Inequalities Ex.2 Solve each inequality. Check your solution. a -1 1a a a - Step: Plug x value back in to original question and check answer. -2 & -3 Solving Inequalities What happens if you multiply or divide an inequality by a negative number? Two things: 1) You do the multiply or divide by the negative number Ex. -2x < to get x < ) You FLIP THE INEQUALITY SIGN SO, x < -2.5 goes to x > CH-2 & -3 Solving Inequalities -2: Evens : ODDS, 1-23 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. Step: Plug x value back in to original question and check answer

4 Ex.1 Solve each inequality. 2(3+3g) 2g + 1 PEMDAS starts it off 6 + 6g 2g + 1-2g -2g 6 + g g +8 Step1: Get x term(s) together on one side. Step3: Get x alone using opposite functions. Step: Plug x value back in to original question and check answer. g 2 13 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 1 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 15 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 r r You can buy NO MORE THAN 8 RIDES 16

5 CH- Solving Multi-Step Inequalities Evens 2-20 CH-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: - < t+2 < Nothing is different than before! You still want to isolate your variable, using your recipe CH-5 Compound Inequalities Ex.1 Solve each inequality. - < t+2 < Steps 1&2 are done - < t+2 < treat it like 2 eqns minus 2 from ALL sides -6 < t < 2 answer has 2 parts Graph it on a number line to see if this result makes sense Step1: Get x term(s) together on one side. Step3: Get x alone using opposite functions. Step: Plug x value back in to original question and check answer. CH-5 Compound Inequalities Odds 1-15 &

6 CH-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 When you have inequalities, it is almost the same, except you have to switch the </> sign for an inequality and make the number negative, to get answers. It is easiest to just break out two equations and solve for the two answers. 21 CH-6 Absolute Value Equations and Inequalities Ex.2 Solve each inequality. 3c-6 3 First, to get rid of Absolute Value bars, rewrite as two separate equations. 3c c-6-3 Step1: Get x term(s) together on one side. Step3: Get x alone using opposite functions. Step: Plug x value back in to original question and check answer. Now solve each equation and combine into one answer, if possible 22 CH-6 Absolute Value Equations and Inequalities 3c c c 9 3c c 3 c +1 c +1 or c 3 Step: Plug x value back in to original question and check answer. CH-6 Absolute Value Equations and Inequalities Evens,