Inequalities in One Variable

Drink at least six glasses of water a day

Store milk at a temperature below 40 degrees F

Eat snacks fewer than 20 calories

Spend at least $10 for a giftAll are examples of inequalities in everyday life.

You will analyze situations involving inequalities in one variable to find and graph their solutions.

An inequality is a statement that one quantity is less than or greater than another.

Less than < Less than or equal to

Greater than

> Greater than or equal to

Everyday phrase Translation Inequality

At least six glasses The number of glasses is greater than or equal to 6

Below 40 degrees The temperature is less than 40o

Fewer than 20 calories

The number of calories is less than 20

At most $10 The price of the gift is less than or equal to $10

Between 35o and 120o 35o is less than the temperature and the temperature is less than 120o

6g

40t

20c

10p

35 120t

You will act out operations on a number line.

Choose ◦ An announcer◦ A recorder◦ Two walkers

The two walkers make a number line from -10 to 10 on the floor.

The announcer and recorder use the Toe the Line table.

The announcer calls out the operation for Walker A and B.

The walkers perform the operations on their numbers by walking to the resulting value on the number line.

The recorded longs the position of each walker after each operation.

For a trial act out the first operation in the table. ◦ Walker A should stand at position 2 and Walker B

should be at position 4. ◦ Enter the inequality sign that describes the relative

position of walker A and B on the number line. (<)

Call out the operations. After the walkers calculate their new

position, record their new positions on the table

Discuss together which inequality sign should be placed between the positions.

What happens to the walkers’ relative positions on the number line ◦ when the operation adds or subtracts a positive

number?◦ Adds or subtracts a negative number?◦ Does anything happen to the direction of the

inequality symbol? What happens to the walkers’ relative positions

on the number line when the operation multiplies or divides by a positive number?◦ Does anything happen to the direction of the

inequality symbol?

What happens to the walkers’ relative positions on the number line when the operation multiplies or divides by a negative number?◦ Does anything happen to the direction of the

inequality symbol? Which operation on an inequality reverse the

inequality symbol? Does it make any difference which numbers

you use? (Fractions, decimals, and integers) Check your findings about the effects of adding,

subtracting, multiplying, and dividing by the same number on both sides of an inequality by creating your own table of operations and walker positions.

Erin says, “I lose 15 minutes of sleep every time the dog barks. Last night I got less than 5 hours of sleep. I usually sleep 8 hours.” Find the number of times Erin woke up.Let x = the number of times Erin woke up.

The number of hours Erin slept is 8 hours, minus 15 min (1/4 hour) times x (the number of times she woke up). This total is less than 5 hours.

8 0.25 5x

8 0.25 5x

8 0.25 8 5 8x Subtracting 5 from both sides

Simplifying 0.25 3x

Dividing both sides by -0.25 12x

The dog woke her up more than 12 times. However, Erin can only wake up a whole number of times so the solution might be more accurately written as x>12, where x is whole number.

In example B, the inequality 8-0.25x<5 was written to represent the situation where Erin slept less than 5 hours, and her sleep time was 8 hours minus 0.25 hour for each time the dog barked. However, Erin can’t sleep less than 0 hours, so a more accurate statement would be the compound inequality 0≤8-0.25x≤5. You can solve a compound inequality in the same way you solved other inequalities; you just need to make sure you do the same operation to all three parts. Solve this inequality for x and graph the solution.

0 8 8 0.25 8 5 8x

0 8 0.25 5x

Subtracting 5 from all three parts

Simplifying 8 0.25 3x

Dividing all three parts by -0.25 32 12x

You learned to write and solve one-variable inequalities and interpret the results based on real-world situations.

You graphed solutions to one-variable inequalities on a number line.

You interpreted an interval graphed on a number line as an inequality statement.

You learned the sign-changed rule for multiplying and dividing both sides of a one-variable inequality by a negative number.