2. Inequalities and Their Graphs
Section 3-1 Part 2

3. Goals
Goal
To write and graph inequalities.

4. Vocabulary
None

5. 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.

6. 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

7. 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.

8. 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

9. Example: Graphing Inequalities
Graph each inequality.
Draw a solid circle at .
A. m ≥
Shade all the numbers greater than and draw an arrow pointing to the right. 1 – 2 3
B. t < 5(–1 + 3)
Simplify.
t < 5(–1 + 3)
t < 5(2)
t < 10
Draw an empty circle at 10.
Shade all the numbers less than 10 and draw an arrow pointing to the left. –4 –2 2 4 6 8 10 12 –6 –8

10. Graph each inequality. C. –1 > y or y < - 1
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.
–3 –2 –

11. Your Turn: Graph each inequality. A. n < 3 B. a ≥ –4
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.
–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 –

12. Your Turn: Graph each inequality. a. c > 2.5 b. 22 – 4 ≥ w
Draw an empty circle at 2.5.
Shade in all the numbers greater than 2.5 and draw an arrow pointing to the right.
a. c > 2.5 –4 –3 –2 –1 1 2 3 4 5 6
2.5
b. 22 – 4 ≥ w
Draw a solid circle at 0.
22 – 4 ≥ w
Shade in all numbers less than 0 and draw an arrow pointing to the left.
4 – 4 ≥ w
0 ≥ w or w ≤ 0 –4 –3 –2 –1 1 2 3 4 5 6
c. m ≤ –3
Draw a solid circle at –3. –4 –2 2 4 6 8 10 12 –6 –8 −3
Shade in all numbers less than –3 and draw an arrow pointing to the left.

13. 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 ≥.

14. Your Turn: Write the inequality shown by the graph.
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 <.
x < 2.5

15. Your Turn: Write the inequality shown by the graph. x >-2
– 4 – 3 – 2 –
| | | | | | | | |

16. Your Turn: Write the inequality shown by the graph. y ≤ 1
– 4 – 3 – 2 –
| | | | | | | | |

17. “No more than” means “less than or equal to.”
Reading Math
“No more than” means “less than or equal to.”
“At least” means “greater than or equal to”.

18. 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
is at least
85°F t ≥ 85
Draw a solid circle at 85. Shade all numbers greater than 85 and draw an arrow pointing to the right.
t  85 75 80 85 90 70

19. 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
$8.50 w ≥
8.50
w ≥ 8.5 4 6 8 10 12 −2 2 14 16 18
8.5

20. 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!