Introduction to Inequalities

Introduction to Inequalities

SOL 6.14a

Why do inequalities matter?

This truck is demonstrating an inequality FAILURE!

We use inequalities often in practial situations without thinking about it.

Think about what you already know about inequalities:

• < is the symbol for "less than"

• > is the symbol for "greater than"

Inequality Basics

• We use inequality symbols to show that expressions are not always equal.  

• We can use inequalities to represent practical situations.

• You should already be familiar with the following symbols:

  < (less than)

  > (greater than)

Multiple Choice

What does the symbol < mean?

• Equal to

• Less than

• Greater than

Multiple Choice

What does the symbol > mean?

• Equal to

• Less than

• Greater than

Multiple Choice

Which of the following is true?

• 0 > 1

• 1 < 2

• 3 > 4

• 5 < 2

Comparing Integers Recap

• Think about when we compared integers.

• Remember that the numbers to the left are smaller/less, i.e. "lefter is lesser."

• Remember that numbers get larger/bigger/greater as you go to the right.

Comparing Integers Recap

• Numbers to the left are less. Numbers to the right are greater.

• P < Q

• S > R

• T < V

• U > Q

Multiple Choice

Which of the following is true?

• -6 > 0

• -4 > 3

• 5 > -2

• -3 > 1

Multiple Choice

Which of the following is true?

• -6 < -1

• 3 < -4

• 0 < -5

• 4 < 1

Inequality Symbols to Know

• In addition to < (less than) and > (greater than), we will also use the following symbols:

  
  

• ≤ (less than or equal to)

• ≥ (greater than or equal to)

Here are some other key words you may see for inequalities in practical situations:

Multiple Choice

What does the symbol < mean?

• less than

• less than or equal to

• greater than

• greater than or equal to

Multiple Choice

What does the symbol > mean?

• less than

• less than or equal to

• greater than

• greater than or equal to

Graphing an Inequality

• Graphing an inequality looks VERY different from plotting a single integer on a number line.

• In the example shown, Mrs. Haley has only one candle. We plot one point on 1 to represent 1 candle.

• Mrs. Nester, however, has at least 20 candles. She might have 20, or 21, or 22, or 23, or 50 or 100.....To show this, we have a closed circle on 20 and an arrow pointing to the right toward possible answers.

Things to know about “less than” <

• We use < to identify situations when something is less than a given value.

• Based on the menu shown, what are some items with prices that are less than $4?  Fries, for example, are less than $4 at $1.49.

• We can represent this with the inequality x < 4.

Multiple Choice

Which price is less than $4.00?

• $5.49

• $4.19

• $6.29

• $3.29

Graphing x < 4 on a number line

We use an open circle to show that $4 items are not included in our answers.

The numbers covered by the arrow are our possible answers or part of our “solution set.” Possible answers are prices less than $4 such as $1.49 or $3.99.

Multiple Choice

Which of the following is true?

• $5.49 < $4.00

• $6.29 < $4.00

• $4.89 < $4.00

• $1.79 < $4.00

Things to know about “less than or equal to” ≤

• We use ≤ to identify situations when something is less than or equal to a given value.

• Based on the Speedy Checkout sign shown, how many items could you purchase and still use this register?  

• We can write an inequality to represent this situation: x ≤ 20.

• The inequality shows that customers with 20 items OR less can use the register.

Multiple Select

If the customers followed the rules, which TWO customers are allowed to use the "20 items or less register" at the store?

• The customer with 50 items.

• The customer with 28 items.

• The customer with 20 items.

• The customer with 10 items.

Graphing x ≤ 20 on a number line

• We use a closed circle to show that 20 items is a possible answer.

• The numbers covered by the arrow as well as the closed circle are our possible answers or part of our “solution set”

• Recall that the sign said that 20 items OR less could use the check out.

Multiple Select

If x ≤ 20 or customer with "20 items or less" are allowed to use the express checkout, select all of the customers who are allowed to use the check-out (assuming they are following the rules).

• A customer with 20 items.

• A customer with 2 items.

• A customer with 18 items.

• A customer with 65 items.

Things to know about “greater than” >

• We use > to identify situations when something is greater than a given value.

• Based on the ad shown, how much could you spend to qualify for free shipping?

• We can write an in inequality to represent this:  x > 50

• The inequality shows that customers who spend more than $50 qualify for free shipping.

Graphing the inequality x > 50 on a number line

• We use an open circle to show that you will not qualify for free shipping if you spend $50.

• The numbers covered by the arrow are our possible answers or part of our “solution set.”

• Recall that customers must spend OVER $50 ($51, $52. $100, $500, etc.) to qualify for free shipping in the example on the previous slide.

Multiple Select

If customers who spend over $50 qualify for free shipping, which TWO customers will get free shipping?

• The customer who spent $52.

• The customer who spent $25.

• The customer who spent $120.

• The customer who spent $50.

Things to know about “greater than or equal to” >

• We use ≥ to identify situations when something is greater than or equal to a given value.

• Based on the water slide sign shown, how tall could you be to ride the water slide?   x ≥ 48

• The inequality shows that customers who are 48 inches tall OR taller may ride the water slide.

Graphing the inequality x ≥ 48 on a number line

• We use a closed circle to show that 48 inches is a possible answer

• The numbers covered by the arrow as well as the closed circle are our possible answers or part of our “solution set.”

• This shows that a person who is 48 inches tall or taller can ride the water slide. You could be 48.5 inches, 49 inches, 50 inches, etc.

Multiple Select

If you must be at least 48 inches tall to ride the water slide, which children will be allowed to ride it? Select all that are true.

• Lillian is 36 inches tall.

• Logan is 48 inches tall.

• Owen is 51 inches tall.

• Kinsley is 60 inches tall.

Recap: Inequalities at a Glance

< or > are graphed using open circles

< or > are graphed using closed circles

Multiple Choice

Inequalities using < or > are graphed with

• an open circle.

• a closed circle.

Multiple Choice

Inequalities using < or > are graphed with

• an open circle.

• a closed circle.