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

Welcome to Math 6

“Write an Inequality to Represent a

Real-World Problem”

Objectives1. Write an inequality of the form x > c

or x < c to represent a constraint or condition in a real-world or mathematical problem.

2. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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.

What is an INEQUALITY?

An inequality expresses a relationship between expressions.

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.

NOTE: Multiple relationships can be shown using inequalities

as in: 5 ≤ b ≤ 8.

We can write inequalities to represent real-world and mathematical situations.  

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.

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.

Situation Mathematical InequalitySpeed limit Legal speed on the highway

≤ 65 miles per hourCredit 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

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.

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?

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

Example 1 continued

20,000 + 12,000 + c ≤ 65,00032,000 + c ≤ 65,000

– 32,000 – 32,000c ≤ 33,000

Example 1 continued

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.

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

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: 

Many problems will not explicitly use words like “at least” or “is less than.”

  Sometimes you will have to reason it through.

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 ≤.

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

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

B. less than or equal to fifteen minutes

C.greater than fifty minutes

D. less than five minutes

 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.

These symbols do not imply

an action in the way

that operation signs might.

They are symbols for expressing relationships between expressions.

What does it mean when an equation or inequality is true?

An equation is true when it states a correct relationship.

For example,

4a+ 3 + 2a= 6a+ 3 This is a true statement because both expressions represent the same quantity.

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.

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?”

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

Guided Practice

+ BonusModeling Solutions on a Number Line

Guided Practice

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

Guided Practice

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

Guided Practice

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

Guided Practice

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

ConclusionWe 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.

Assignments

Complete the practice exercises attached to the lesson