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Unit 4: Solving Inequalities Inequalities are used to ___________________ one amount to another. There are _____ symbols that can show inequality and we read them from ________________ to ____________ > < > < x > 4 means that x is ________________ than 4. We don’t know what number it is, but it must be something bigger than 4. Use your inequality symbols to translate these problems from words into math. X is less than 4: ______________ 5 is greater than or equal to y: _______________ 3 is greater than w: ______________ b is less than or equal to 8: ______________ Graphing Inequalities on the Number Line: Step 1: On your number line, find the number in your inequality and put a _______ where it is. Step 2: Leave the circle empty if your inequality is ____ or _____. Fill in the circle if your inequality is_____ or________. Step 3: Shade your number line based on what the inequality says… if your variable is greater or greater than or equal to a number, you shade to the __________, where the numbers are ___________. If your variable is less than or less than or equal to a number, you shade to the ____________, where the numbers are ________________. Example 1: x > 4 Step 1: Put a circle on the number line at the _________. -5 -4 -3 -2 -1 0 1 2 3 4 5 Step 2: Leave the circle empty if your inequality is > or <. Fill in the circle if your inequality is > or <. -5 -4 -3 -2 -1 0 1 2 3 4 5 Step 3: Shade your number line based on what the inequality says… where are the numbers greater than 4? -5 -4 -3 -2 -1 0 1 2 3 4 5 Example 2: 3 < x Step 1: Put a circle on the number line at the ______. -5 -4 -3 -2 -1 0 1 2 3 4 5 Step 2: Leave your circle empty if your inequality is > or <. Fill in the circle if your inequality is > or <. -5 -4 -3 -2 -1 0 1 2 3 4 5 Step 3: Shade your number line based on what the inequality says… where are the numbers greater than or equal to 3? -5 -4 -3 -2 -1 0 1 2 3 4 5 You try! Graph these on the number line. 1. x > 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 2. -1 < x 3. 4 > x 4. x < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 When you have a negative x, you need to turn it into a positive! How do we get rid of a negative in front of an x? _________________________________________________ When we divide by -1, our inequality sign ______________. If it was >, it becomes < and if it was <, it becomes >. So if it was >, it becomes < and if it was _____ it becomes _____. Example 3: -x > 2 You try! 1. 4 < -x 2. –x > 3 3. –x < -1 4. 5 > -x -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Tuesday - Solving One Step Inequalities For one step equations, we only need to use ___________________ _________________ one time to ________________ the variable. Use what you learned yesterday to try these! Solve: 1. x + 3 > 7 2. 3x ≤ 18 3. -6n < - 12 4. 4 + x > 2 5. Z – 8 ≥ -7 Now, solve the inequality and graph it on the number line! 1. -8x > 32 2. x + 9 ≤ 15 Solving Two Step Inequalities Solving two step inequalities is almost exactly the same as solving two step __________________________. REMEMBER: The goal is ALWAYS TO _____________________ the variable OR get it by _______________ Step 1: Do the inverse operation of ___________________________ or __________________________ first. Step 2: Do the inverse operation of ___________________________ or ______________________________ next. If you are multiplying or dividing by a _____________________ number, you must _________________ the inequality sign. Step 3: __________________ for the variable. Step 4: Plug your answer back in. Examples: x 1. 2x – 9 < 5 2. + 7 > 3 3. -3x – 10 < -16 2 More Practice 1. 7x – 12 ≤ 23 2. x +7≥3 6 4. -3x – 10 < -16 5. 30 > -x + 18 7. 30 > 4 - x 8. 9x + 5 ≤ 43 Now solve and graph! 3. 5x + 6 < -21 6. x + 6 > 23 4 9. -7x + 3 ≥ 14 1. 4 – 9c ≥ 13 2. -18 < 5m – 3 3. x + 4 < 6 2 4. 4d + 7 ≥ 23 5. -10 ≤ 5m – 3 6. -5x + 9 > -14 Challenge : Solve each (do not graph) 1. -2(h+2) < -14 2. -3t + 7 ≤ -5t + 9 3. -4(x -5) > 6(x + 3) Wednesday – Absolute Value Inequalities There are two types of inequalities with absolute values. Type 1: “Less Than” Inequalities Remember that absolute value is the distance from a number to ________ on the number line. Let’s think about x < 3. The problem is saying that any answer to x needs to be less than ____ units from 0 on the number line. Where on the number line could x fall? Generally speaking, if an absolute value equation is _______________ another value, the answers will all fall BETWEEN two end points. It will follow the pattern below: a b -b < a < b Example: 6x 12 Step 1. Rewrite the equation: < 6x < Step 2: Split up the two inequalities < 6x and 6x < Step 3: Solve each inequality as if it were an equation. ________________________________________________ You try! 5p 20 2m 40 4g 1 18 6 3y 31 Type 2: “Greater Than” Inequalities Let’s think about x 2 . The problem is saying that any answer to x needs to be more than ____ units from 0 on the number line. Where on the number line could x fall? When an absolute value equation is ____ _________________ another value, it can go on infinitely in both directions. All of the answers will be ___ ____________ one number and __ ______________ its opposite. It will follow the pattern below: a b a > b or a < -b Example: 9x 36 Step 1. Rewrite the equation: 9x > or 2: Solve each inequality as if it were an equation. Step ________________________________________________ 9x < You try! 6p 30 8w 40 4 2y 12 Thursday - Multi-Step Inequalities The process for solving inequalities is the __________ as solving equations with ONE exception: when you divide by a ___________________ you must __________ the inequality symbol! Examples: 1.) 4(x+ 3) ≤ 24 2.) -2(x – 8) > 32 3.) 4x – 5 ≤ -6x + 15 4.) 8x + 6 < 2x - 12 You Try! 1.) 5(x+ 2) ≥ 30 2.) -3(x + 3) < 9 3.) 8x – 4 ≥ -10x + 32 4.) 4x + 2 > 10x - 22 Challenge Problems: 5.) 4 x 2 4 7.) 6 x 4 6 6.) 5 x 10 15 Friday - Compound Inequalities Vocabulary Compound Inequality: Solving a Compound Inequality with AND - What will it look like? _______________ < _______________ < ____________________ Step 1 Write the _________________ inequality 2 Isolate the ____________ between the two inequality symbols by first __________ or __________ 3 __________ or __________ the number with x to both numbers on the outside of the inequalities. Example # 1 Example # 2 Example # 3 -2 < 3x – 8 < 10 -3 < 4x – 9 < 11 4x - 24 < 30 < 9x - 15 **** Remember to switch both inequality symbols if you multiplied or divided by a negative number!!! 4 Write your solution as one inequality _____ <x<_____ 5 Plot both inequalities on the ___________ number line X>2 0 AND X<6 2 6 Guided Practice Problems – Solve the following Compound Inequalities and Plot on a Number Line 1. -5 < 3x + 4 < 19 2. – 4 < 2 + x < 1 3. -25 < 5x < -20 Solving a Compound Inequality with OR - What will it look like? _______________ OR ____________________ Step 1 Write the _________ inequality 2 Solve each inequality _______________ for x Example # 1 Example # 2 2x + 7 < 3 or 5x + 5 > 10 3x + 8 > 17 or 2x+5< 7 **** Remember to switch the inequality symbol if you multiplied or divided by a negative number!!! 3 Write your solution as _________ inequalities with OR in the middle x < -2 6 Plot both inequalities on the __________ number line -2 x>1 1 Example # 3 -3x -7>8 or -2x -11< -31 Guided Practice Problems – Solve the following Compound Inequalities and Plot on a Number Line 6+2x > 20 or 8 + x < 0 3x + 1 < 4 or 2x – 5 > 7 -2x> 6 or 2x + 1 > 5 Monday - Inequality Word Problems Why do we have inequalities? Think about the statements below: 1) John has 5 dollars. ________ 2) John has at least 5 dollars. __________ 3) John has less than 5 dollars._______ Why are they different? Why are they the same? How could you write each one? When we know an exact amount, we use an ___________________ sign to show that the two sides are the ___________________. When we don’t know an exact amount, we use ________________________ to compare sides. Inequality Symbol > ≥ < ≤ Words Associated with Symbol Example Problems: 1) Yolando needs more than $5,000 to start her own business. 2) The speed limit is 80 MPH. a) Write an inequality and b) draw it on the number line. 3) Yellow Cab charges a $1.25 flat rate and an additional $0.30 per mile traveled. Jared has no more than $10.00 to spend on his cab ride. a) Write an inequality to represent Jared’s situation. b) How many miles can Jared travel without going over his budget? 4) Domino’s sells medium 2-topping pizzas (p) for $8.00 and 2-liter sodas for $2.00 (s). The company wants to make at least $80.00 per night. Write an inequality that expresses this situation. 5) The minimum age for the concert was 14 years old. You Try! 1) The school bus fits no more than 40 students. a) Write an inequality to describe this and b) graph it. 2) Sarah needs more than 6 hours of sleep, but at most 9 hours. a) Write an inequality to describe this and b) graph it. 3) Jacob has a lawn-mowing business. He makes $100 per week and $35 per lawn he mows. He needs to make at least $240 per week to save up for his new car. a) Write an inequality to represent Jacob’s situation. b) How many lawns must he mow to make $240 per week? 4) To ride a rollercoaster, a person must be a minimum of 48” tall, but cannot exceed 80”. 5) The maximum weight limit for the bridge is 6000 pounds