Outcome 3 Inequalities

What are inequalities? Break it down… Do you already know part of the word? What does that mean? How is this different?

Inequalities > Greater than < Less than An inequality is basically like an equation, but instead of having an equal sign, it has an inequality sign Examples of inequality signs: > Greater than < Less than ≥ Greater than or equal to (line under) ≤ Less than or equal to (line under)

Plotting inequalities on a number line x > 7 x < -3 7 < x -3 > x

Plotting inequalities on a number line x ≤ 7 x ≥ -3 7 ≥ x -3 ≤ x

What’s with the dots? If an inequality is only > or < , then the dot is NOT filled in If an inequality is ≤ or ≥ , then the dot IS filled in (because it includes that number!)

Solving One-Step Inequalities 2 3 4 5 6 7 8

Solving One-Step Inequalities Objective: To solve one-step inequalities using properties for inequalities

Solving One-Step Inequalities What are the Properties for Inequalities? Are they the same as for regular equations?

Solving One-Step Inequalities Addition Property for Inequalities X - 5 < 7 +5 +5 X < 12

Solving One-Step Inequalities Subtraction Property for Inequalities X +15 < 25 - 15 -15 X < 10

Solving One-Step Inequalities Multiplication Property for Inequalities

Solving One-Step Inequalities Division Property for Inequalities Divide both sides by the coefficient of x

The sign changes direction when you divide or multiply by a negative number!!!

Solving One-Step Inequalities 1

Solving One-Step Inequalities 2

Solving One-Step Inequalities 3

Solving One-Step Inequalities 4

Write & solve an inequality 4 times a number minus 7 is less than or equal to 21 4 plus 2 times a number x is greater than 14 minus 6 times a number x

Writing an inequality I’m saving up for a new TV (a huge one). Right now I’ve got $500 set aside and I want to put away $50 per month until I save at least $1,000 to spend on it. Mr. Bantz went to the store and wanted to spend at most $22. He needed to buy 12 rulers for a project and 3 large binders.

Mr. Bantz wants to re-enact the bonfire during homecoming week Mr. Bantz wants to re-enact the bonfire during homecoming week. He wants to have a pie eating contest and a hula-hoop contest and spend at most $60. Pies cost $6 each and Hula-hoops $4 each. Write an inequality for this situation. Is it possible that he bought 5 pies and 6 hula-hoops? Is it possible that he bought 4 pies and 10 hula-hoops?

Mr. Still owns a company that sells Metallica memorabilia Mr. Still owns a company that sells Metallica memorabilia. He sells old concert t-shirts for $15 each and old ticket stubs for $10 each. Brett wants to make at least $300 profit. Write an inequality for this situation. Is it possible that he sold 10 shirts and 10 stubs? Is it possible that he sold 12 shirts and 15 stubs?

Yorkville Clothing Co. sells sweatshirts and t-shirts Yorkville Clothing Co. sells sweatshirts and t-shirts. T-shirts sell for $12 each and sweatshirts sell for $16 each. The company made at least $296 Write an inequality for this situation. Is it possible that the company sold 10 sweatshirts and 10 t-shirts? Is it possible that the company sold 15 sweatshirts and 5 t-shirts?

Mr. Muth owns a company that sells shirts with his photo on the front, he also sells hats with his picture on it. He sells the shirts for $20 each and hats for $15 each. He knows he has made at least $200. Write an inequality for this situation. Is it possible that the company sold 10 shirts and 10 hats? Is it possible that the company sold 5 shirts and 8 hats?

Compound Inequalities

What does the word compound mean? A compound inequality is two inequalities joined together by the word “and” or by the word “or”.

To understand this a bit more lets examine a situation. Given these two options which would you choose? Why? Your homework is to write a paper and do 15 math problems. Your homework is to write a paper or do 15 math problems. http://www.youtube.com/watch?v=773mIAk9Y2Q&safe=active

So what does each mean for us? In option 1, the and means you have to satisfy both requirements. Both the paper and problems must be done. The boss must be large AND in charge. In option 2, the or means you have to satisfy only one requirement. Either the paper or the problems must be done. The boss is either large OR in charge.

Lets examine an AND inequality! Lets say you were given the compound inequality: 𝒙≥𝟓 𝒂𝒏𝒅 𝒙<𝟏𝟎 How would that look? For AND we need to satisfy both, so where does it satisfy both of the inequalities?

Lets examine an OR inequality! Lets say you were given the compound inequality: 𝒙<𝟐 𝒐𝒓 𝒙>𝟔 How would that look? For OR we need to satisfy only one, so where does it satisfy at least one of the inequalities?

True Compound inequalities Lets examine : −3<𝑥≤5 This is a compound inequality, it contains 2 inequalities…. What are they? −3<𝑥 and 𝑥≤5 Any time we have a compound inequality in this form we assume that it is an AND inequality. So to be clear the only time we have an or inequality is if it says… OR!

Lets say you were given the compound inequality: Now for some fun! Lets say you were given the compound inequality: −7<2𝑥+1≤5 What are your two inequalities? −7<2𝑥+1 and 2𝑥+1≤5 Now solve each: −7<2𝑥+1 2𝑥+1≤5 −8<2𝑥 2𝑥≤4 −4<𝑥 𝑥≤2

Lets say you were given the compound inequality: Now for some fun! Lets say you were given the compound inequality: −4< 𝑥 2 −3 ≤1 What are your two inequalities? −4< 𝑥 2 −3 and 𝑥 2 −3 ≤1 Now solve each:

Lets say you were given the compound inequality: Now for some fun! Lets say you were given the compound inequality: 4𝑥+2>18 𝑜𝑟 2𝑥−4≤0 What are your two inequalities? 4𝑥+2>18 𝑜𝑟 2𝑥−4≤0 Now solve each:

Lets say you were given the compound inequality: Now for some fun! Lets say you were given the compound inequality: 10−2𝑥<2 𝑜𝑟 5𝑥+10≤20 What are your two inequalities? Now solve each:

No solution!!!!! OH SNAP! Changing it up… Solve and graph −10>−4𝑥+2>10 What are your two inequalities? −10>−4𝑥+2 and−4𝑥+2>10 Now solve each: We should end up with 3<𝑥 𝑎𝑛𝑑 𝑥<−2 Now graph it… This is an AND so where do they overlap? Never? What does that mean? No solution!!!!!

Compound inequality word problems

Write a compound inequality for each situation The roller coaster vertical velocity at Six Flags Great America has height limitations. To ride V2, you must be at least 48 inches tall, and less than 78 inches tall.

Mr. Muth went for a run and ran somewhere between 20 and 24 miles Mr. Muth went for a run and ran somewhere between 20 and 24 miles. If Mr. Muth is running at a speed of 8 miles per hour, for what range of time was he running?

Erin makes $30 per hour. She is saving up money for a new moped and going into work this week already had $200 saved up. Find the number of hours Erin must work if the moped costs somewhere in the range of $1,460 to $2,000.

Mr. Bantz has created the super awesome club Mr. Bantz has created the super awesome club. To join the super awesome club a person must meet some criteria. The members age multiplied by 3 and adding on one must be less than 76 or the members age divided by 2 and subtracting 3 must be greater than or equal to 12. What ages can join the club?

Graphing Linear Inequalities

Linear Inequalities WS http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Graphing%20Linear%20Inequalities.pdf

Summary > or < is a dashed line ≥ or ≤ is a solid line Plot them exactly like you would an equation (make sure your line is dashed or solid Pick a point, see if it works, then shade in the appropriate side of the inequality

Graphing Systems of Linear Inequalities

Systems of Linear Inequalities WS http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Systems%20of%20Inequalities.pdf

Summary Graph each inequality separately (make sure your lines are correctly dashed or solid) After graphing each line, pick a point, see if it works, and lightly shade in the correct side After graphing both inequalities, darkly shade in the area where you overlapped shading lightly To check, you can pick a point in your dark shaded area and make sure it works in both inequalities