©2013 Judo Math Inc.

INEQUALITIES

©2013 Judo Math Inc.

7th grade

Algebra/Proportions Discipline: Black Belt Training

Welcome to the Black Belt – All about inequalities After completing the last discipline you might be thinking that math is all about “equals.” We solved lots of equations to figure out what the variable would EQUAL. In this discipline, we are going to completely turn this upside down. In fact, we’re not going to figure out what anything EQUALS…. But we will figure out what it’s bigger than or smaller than. What am I talking about?! Inequalities! You may remember your teachers in elementary school teaching you about the alligator mouths < and >. Remember the alligator mouth always pointed to the larger quantity… because obviously an alligator would always want to eat the bigger amount. For example…

4>3 5<10 17>12 if x>10 then x can be 11, 12, 13, 14, 15, …

Someday you might do something crazy like prove Cauchy’s very famous inequality:

But for now, we will just try to not get eaten by the alligator mouth.

Good Luck Grasshopper.

Order of Mastery: All about inequalities 1. Define inequality

2. Graphing Inequalities 3. Solve and graph one step inequalities (7EE4b) 4. Solve and graph two or more step inequalities (7EE4b)

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1. Define Inequalities An Inequality tells you about the relative size of two values. Mathematical inequalities are around you almost every day, but you probably don’t notice them because they are so familiar. Think about the following situations: speed limits on the highway, minimum payments on credit card bills, number of text messages you can send each month from your cell phone, and the amount of time it will take to get from home to school. All of these can be represented as mathematical inequalities. And, in fact, you use mathematical thinking as you consider these situations on a day-to-day basis.

Situation Mathematical Inequality

Speed limit Legal speed on the highway ≤ 65 miles per hour

Credit card Monthly payment ≥ 10% of your balance in that billing cycle

Text messaging Allowable number of text messages per month ≤ 250

Travel time Time needed to walk from home to school ≥ 18 minutes

When we talk about these situations, we often refer to limits, such as “the speed limit is 65 miles per hour” or “I have a limit of 250 text messages per month.” However, we don’t have to travel at exactly 65 miles per hour on the highway, or send and receive precisely 250 test messages per month—the limit only establishes a boundary for what is allowable. Thinking about these situations as inequalities provides a fuller picture of what is possible. There are 4 very important symbols when working with inequalities… here they are:

Example 1: Alex and Billy have a race, and Billy wins!

What do we know?

We don't know how fast they ran, but we do know that Billy was faster than Alex:

Billy was faster than Alex

We can write that down like this:

b > a

(Where "b" means how fast Billy was, ">" means "greater than", and "a" means how fast Alex was)

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Example 2: Alex plays in the under 15s soccer. How old is Alex?

We don't know exactly how old Alex is, because it doesn't say "equals"

But we do know "less than 15", so we can write:

Age < 15

The small end points to "Age" because the age is smaller than 15.

Example 3: You must be 13 or older to watch certain movies.

Can YOU write an inequality for this scenario?: ________________

If you wrote Age ≥ 13 then you are correct!

Write inequalities for each of these situations:

1. The distance to Philadelphia is greater than 100 miles

2. You must be 18 or older to watch this film.

3. You must be older than 16 to get your drivers license.

4. To pass this class, you need greater than 70%

5. To not gain weight I need to eat less than 1200 calories per day.

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Now you try to come up with 5 other situations from your life that can be modeled by an inequality

(hint; bedtime, curfew, time spend on homework, etc.)

1.

2.

3.

4.

5.

Now practice converting these number sentences into inequalities.

Example sentence: A number minus 4 is greater than 2.

Inequality: x-4>2

1. A number less 5 is greater than 7

2. The sum of 3 and a number is less than or equal to -9.

3. A number plus 3 is greater than or equal to 12.

4. The sum of x and 5 is less than or equal to -2.

5. The sum of a number and 2 more than the number is less than 20.

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6. Five pounds of ground beef cost more than $5.00

7. The sum of x and 3 is greater than or equal to 15.

8. A number times five is greater than fifteen.

9. A number less negative seven is less than or equal to five.

10. The difference between a number and seven is less than zero.

11. The sum of a number and seven is greater than or equal to fifteen.

12. The sum of a number and nine is less than thirty.

13. Jamie’s dog ate her homework (cliché, right?). Anyway, the only part that is left is

her answer to the last problem that says 2x>30. Can you come up with at least

two possible questions that she may have been answering to obtain this answer

on her homework?

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2. Graphing inequalities

In the inequality x < 4, "x" could be any number less than 4.

This not only includes integers, but decimals and fractions, too. We can graph the solution set on a number line.

To graph our solution set we must show that all the numbers less than 4 (but not including the

4) could replace the "x" in x < 4.

We show this by placing an open circle on the 4 and drawing an arrow to the left to show "less than".

Let's graph the solution set for the inequality x ≥ -1.

To graph this solution set we must show that all the numbers greater than -1, including the -

1, could replace the "x".

We show this by placing a closed circle on the -1 and drawing an arrow to the right to show "greater than".

Things to remember...

If the symbol in the inequality is < or >, use an open circle above the number being graphed.

If the symbol in the inequality is ≤ or ≥, use a closed circle above the number being graphed.

If the symbol in the inequality is < or ≤, draw an arrow to the left to show "less than".

If the symbol in the inequality is > or ≥, draw an arrow to the right to show "greater than".

Write an equality from a number line graph

1.

Inequality:

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4. Graph x < 3 on a number line.

5. Graph x ≥ 0 on a number line.

6. Graph x > -2 on a number line.

7. Graph x ≤ 1 on a number line.

2.

3.

Inequality:

Inequality:

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6. Graph x > -5 on a number line.

7. Graph x ≤ 1

2 on a number line.

8. Graph x > -11

2 on a number line.

9. Graph x ≤ 4.2 on a number line.

10. Graph x>71

3 on the number line.

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3. Solve one step inequalities

One-step inequalities (using the symbols <, >, ≤, ≥) are solved the same as one-step equations.

For example: x + 18 = 40 would be solved by taking 18 away from both sides to get "x" by itself

The inequality x + 18 < 40

would be solved by taking 18 away from both sides to get "x" by itself resulting in the answer: x<22

Cake, right? Well there is one little hiccup… it comes when you need to multiply or divide by a negative number (negative numbers are just so pesky, aren’t they?) Check out this picture then fill in the blank below…

When dividing or multiplying by a negative number, you must always _______________ the inequality sign…. In other words, if the alligator mouth is opening to the right, turn it around to open to the left. Weird huh? Why reverse the sign? Check out this equation solving to see why…

-4y < -36 Add 4y

0 < 4y - 36 Add 36

36 < 4y Divide by 4

9 < y Reverse the whole inequality

y > 9

By avoiding multiplication by a negative number, I avoided the need to

reverse signs until the end and so I can see that if I had just divided by -4

in the beginning, I would have had to flip the sign!

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Here is another example of solving an inequality. For each problem, fill in the blank to the right to

explain what happened in that step.

-2x - 10 < 2

-2x - 10 + 10 < 2 + 10 I added 10 to both sides to cancel out the 10 on the right (inverse operations)

-2x < 12 The 10’s canceled and I combined the 2 and the 10.

x > -6 I divided both sides by negative 2 (to undo the multiplication) and flipped the inequality sign around

Now you make comments in the blanks on this one…

-2x + 15 < 3

-2x + 15 - 15 < 3 – 15 _______________________________

-2x < -12 ___________________________________

x > 6 _________________________________

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4. Solve and graphing two or more step inequalities

Now that you are a master of the basics of inequalities, you are ready to go big! Just like equations, inequalities can get a lot more complicated to write and solve. In the last section, each equation required only one-step to solve. Now we come upon situations where inequalities that require 2 or more steps arise… for example…

Write an inequality and graph its solution on a number line for each of these situations: Bake Sale Problem: The school bake sale needs to make at least $200. If the receive a donation of $60 and then each cake is sold for $10 how many cakes should they sell to beat their goal? Using arrows to point, explain each part of your equation.

2

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Savings Account: Bob has $700 in his savings account at the beginning of June. If he wants to keep at least $300 in the account for school next September and he withdraws $25 every week for spending money, how many weeks can he withdraw money?

Big Trucks!: An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?

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Running Shoes: Erika has found three pairs of running sneakers that she likes, costing $150, $159, and $179. She has saved $31 already, and she has a job where she earns $8.50 per hour. How many hours will she have to work in order to afford any of these sneakers? A story about Jack… Jack earns $75 a day and $15 per hour for overtime. In a month, he earned at least $700 for the five weekdays of any week. What is the minimum number of hours he worked overtime for any week of that month?

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More practice… solve each inequality and graph the solution.

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