1. “Write an Inequality to Represent a Real-World Problem”
Welcome to Math 6
“Write an Inequality
to Represent a
Real-World Problem”

2. Objectives
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.
Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

3. Connector
Inequalities are used all the time in the world around us. Figuring out how to interpret the language of inequalities is an important step toward learning how to solve them in everyday contexts.

4. What is an INEQUALITY? An inequality expresses a relationship between expressions.

5. These relationships come in five forms:
A ≠ (is not equal) B. A > (is greater than) B. A ≥ (is greater than or equal to) B. A < (is less than) B. A ≤ (is less than or equal to) B.

6. NOTE: Multiple relationships can be shown using inequalities as in: 5 ≤ b ≤ 8.
We can write inequalities to represent real-world and mathematical situations.

7. We see mathematical inequalities often but we may not notice them because they are so familiar.
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.

8. 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.

9. 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

10. Example 1
An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds.

11. Example 1 continued
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?

12. cab weight trailer weight cargo weight ≤ allowable weight
Example 1 continued
We can represent the situation using the following inequality, where c is the weight (in pounds) of the truck’s cargo:
cab weight +
trailer weight
cargo weight ≤
allowable weight
20,000
12,000 c
65,000

13. Example 1 continued
20, , c ≤
65,000
32, c
– 32,000 c
33,000

14. Example 1 continued
Solving this inequality for c, we find that c ≤ 33,000. This means that the weight of the cargo in the truck can be anywhere between 0 pounds and 33,000 pounds and the truck will be allowed to cross the bridge.

15. “a is at most b;” or “a is no more than b”
When you are solving or building these inequalities, it is important to know which inequality symbol you should use. Watch for certain phrases that will tip you off:
Phrase
Inequality
“a is more than b”
a > b
“a is at least b”
a ≥ b
“a is less than b”
a < b
“a is at most b;” or “a is no more than b”
a ≤ b

16. Sometimes you will have to reason it through.
Many problems will not explicitly use words like “at least” or “is less than.”   
Sometimes you will have to reason it through.

17.   
For example, in the truck problem that we solved, the “maximum” weight allowed on this bridge was 65,000 pounds. The total weight of the cab, trailer, and cargo had to be “no more than” 65,000.
Since the clue stated “no more than…” it could be “equal to” and it could be “less than,” thus ≤.

18.   
Once we have identified the relationship between the two quantities we can put in the appropriate symbol.

19. A. greater than or equal to fifteen minutes
Example 2
A bus stops at the corner every thirty minutes, on the half hour. The bus is sometimes up to five minutes late. Your watch says it is 3:50 PM. The time you wait for the bus will probably be:
A. greater than or equal to fifteen minutes
less than or equal to fifteen minutes
greater than fifty minutes
D. less than five minutes

20.  Example 3
The class must raise at least $100 to go on the field trip. They have collected $20. Write an inequality to represent the amount of money, m, the class still needs to raise. Represent this inequality on a number line.

21. These symbols do not imply an action in the way
that operation signs might.
They are symbols for expressing relationships between expressions.

22. What does it mean when an equation or inequality is true?
An equation is true when it states a correct relationship.
For example,
4a a= 6a+ 3
This is a true statement because both expressions represent the same quantity.

23. An inequality is true when it states a correct relationship
An inequality is true when it states a correct relationship. For example:
4a + 3 ≠ 6a + 3
This is a true statement because both expressions represent different quantities.

24. How is solving an inequality a process of answering a question?
Solving inequalities is like asking, “For what numbers will this equation or inequality be true?”

25. Free Shipping for all orders $24.99 and over.
What amounts can receive free shipping? An infinite number of values can receive free shipping--anything $24.99 and up. Let “a ” represent free shipping orders, so a> $24.99

26. Modeling Solutions on a Number Line
Guided Practice
+ Bonus
Modeling Solutions on a Number Line

27. Guided Practice
Children 3 years old and older may ride any of the rides in the local amusement park.

28. Guided Practice
Children older than 3 may ride any of the rides in the local amusement part.

29. Guided Practice
Temperatures at Lake Chilly are expected to be 8° C and below on Tuesday.

30. Guided Practice
Temperatures at Lake Chilly are expected to be below 8°C on Tuesday.

31. Conclusion
We can represent a constraint or condition in a real-world or mathematical problem by writing an inequality of the form x > c or x < c. Inequalities in the form x > c or x < c have infinitely many solutions. We can represent solutions of such inequalities on number line diagrams.

32. Assignments
Complete the practice exercises attached to the lesson