Inequalities and
Their Graphs
Section 3-1 Part 2
Goals
Goal
• To write and graph
inequalities.
Vocabulary
• None
Solutions to Inequalities
An inequality like 3 + x < 9 has
too many solutions to list. You
can use a graph on a number
line to show all the solutions.
The solutions are shaded and an arrow shows
that the solutions continue past those shown on
the graph. To show that an endpoint is a solution,
draw a solid circle at the number. To show an
endpoint is not a solution, draw an empty circle.
Graphing Solutions to Inequalities
The symbols < and > indicate a open circle.
This open circle shows that 5 is not a solution.
a>5
The symbols ≤ and ≥ indicate a closed circle.
This closed circle shows that 3 is a solution.
b≤3
Graphing Solutions to Inequalities
Always write the inequality with the variable on the
left side, then the arrow points in the same direction
as the inequality sign.
Rewriting Inequalities
• When you rewrite an inequality to
change which side the variable is
on (reverse the inequality), you
change the direct of the inequality
sign (reverse the inequality sign)
also.
• Example:
– If 4 < x, then x > 4
– If y ≥ -3, then -3 ≤ y
Example: Graphing Inequalities
Graph each inequality.
Draw a solid circle at
A. m ≥
–
0
1
2
3
Shade all the numbers
greater than and draw an
arrow pointing to the right.
3
B. t < 5(–1 + 3)
t < 5(–1 + 3)
Simplify.
Draw an empty circle at
10.
t < 5(2)
t < 10
–8 –6
–4
.
–2
0
2
4
6
8
10 12
Shade all the numbers
less than 10 and draw an
arrow pointing to the left.
Graph each inequality.
C. –1 > y
–3 –2
or y < - 1
–1
0
1
2
3
Rewrite the inequality as y<-1.
Draw an open circle at –1. The
solutions are all values of y
less than –1, so shade the line
to the left of –1.
Your Turn:
Graph each inequality.
Draw an open circle at 3. The
solutions are all values of n less
than 3, so shade the line to the
left of 3.
A. n < 3
–3
–2
–1
0
1
2
3
B. a ≥ –4
Draw a closed circle
at –4. The solutions are all values
greater than –4, so shade to the
right of –4.
–6
–4
–2
0
2
4
6
Your Turn:
Graph each inequality.
Draw an empty circle at 2.5.
a. c > 2.5
Shade in all the numbers
greater than 2.5 and draw an
arrow pointing to the right.
2.5
–4 –3 –2 –1
0
1
2
3
4
5
6
b. 22 – 4 ≥ w
22 – 4 ≥ w
4–4≥w
0 ≥ w or w ≤ 0
–4
–3 –2 –1
0
1
2
Draw a solid circle at 0.
Shade in all numbers less than
0 and draw an arrow pointing
to the left.
3
4
5
6
c. m ≤ –3
Draw a solid circle at –3.
−3
–8 –6 –4
–2
0
2
4
6
8
10 12
Shade in all numbers less
than –3 and draw an arrow
pointing to the left.
Example: Writing an Inequality from
a Graph
Write the inequality shown by each graph.
x<2
Use any variable. The arrow points to the left, so use either
< or ≤. The empty circle at 2 means that 2 is not a solution,
so use <.
x ≥ –0.5
Use any variable. The arrow points to the right, so use
either > or ≥. The solid circle at –0.5 means that –0.5
is a solution, so use ≥.
Your Turn:
Write the inequality shown by the graph.
x < 2.5
Use any variable. The arrow
points to the left, so use either <
or ≤. The empty circle at 2.5
means that 2.5 is not a solution,
so use so use <.
Your Turn:
Write the inequality shown by the graph.
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–4
–3
–2
–1
0
1
2
3
4
x >-2
Your Turn:
Write the inequality shown by the graph.
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–4
–3
–2
–1
0
1
2
3
4
y≤1
Reading Math
“No more than” means “less than or
equal to.”
“At least” means “greater than or
equal to”.
Example: Application
Ray’s dad told him not to turn on the air conditioner unless
the temperature is at least 85°F. Define a variable and write
an inequality for the temperatures at which Ray can turn on
the air conditioner. Graph the solutions.
Let t represent the temperatures at which Ray can turn on
the air conditioner.
Turn on the AC when temperature
≥
t
t  85
70
75
80
is at least
85
90
85°F
85
Draw a solid circle at 85.
Shade all numbers greater
than 85 and draw an arrow
pointing to the right.
Your Turn:
A store’s employees earn at least $8.50 per hour. Define
a variable and write an inequality for the amount the
employees may earn per hour. Graph the solutions.
Let w represent an employee’s wages.
An employee earns
at least
≥
w
w ≥ 8.5
8.5
−2
0
2
4
6
8 10 12 14 16 18
$8.50
8.50
Joke Time
• What has 18 legs and catches flies?
• A baseball team.
• Who stole the soap?
• The robber ducky!
• What happened to the plant on the
windowsill of the classroom?
• It grew square roots!