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 gift
 All 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
g6
Below 40 degrees
The temperature is
less than 40o
t  40
Fewer than 20 calories
The number of
calories is less than 20
c  20
At most $10
The price of the gift is
less than or equal to
$10
p  10
Between 35o and 120o
35o is less than the
temperature and the
temperature is less
than 120o
35  t  120


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  0.25x  5
8  0.25x  5
Subtracting 5 from both sides
Simplifying
Dividing both sides by -0.25
8  0.25x  8  5  8
0.25x  3
x  12
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  0.25x  5
Subtracting 5 from all three parts
Simplifying
Dividing all three parts by -0.25
0  8  8  0.25x  8  5  8
8  0.25x  3
32  x  12




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 onevariable inequality by a negative number.