1. 2-5 Warm Up Problem of the Day Lesson Presentation
Solving Inequalities Containing Integers
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. 2-5 > > < = > <
Course 3
2-5
Solving Inequalities Containing Integers
Warm Up
Compare, Use >, <, or =.
>
1. – –4
> –2
3. – –5
<
4. – –5 =
> –3
6. – –2
<

3. Course 3
2-5
Solving Inequalities Containing Integers
Problem of the Day
Elliot led the race; Gordon was ahead of Wallace. Labonte was directly behind Gordon and directly ahead of Marlin. Who was in last place ?
Wallace

4. 2-5 Learn to solve inequalities with integers.
Course 3
2-5
Solving Inequalities Containing Integers
Learn to solve inequalities with integers.

5. 2-5 Solving Inequalities Containing Integers
Course 3
2-5
Solving Inequalities Containing Integers
When you pour salt on ice, the ice begins to melt. If enough salt is added, the resulting saltwater will have a freezing point of –21°C, which is much less than water’s freezing point of 0°C.
Adding rock salt to the ice lowers the freezing point and helps to freeze the ice cream mixture.
At its freezing point, a substance begins to freeze. To stay frozen, the substance must maintain a temperature that is less than or equal to its freezing point.

6. 2-5 Solving Inequalities Containing Integers
Course 3
2-5
Solving Inequalities Containing Integers
If you add salt to the ice that is at a temperature of –4°C, what must the temperature change be to keep the ice from melting?
This problem can be expressed as the following inequality:
–4 + t  –21
When you add 4 to both sides and solve, you find that if t  –21, the ice will remain frozen.

7. 2-5 Solving Inequalities Containing Integers
Course 3
2-5
Solving Inequalities Containing Integers
The graph of an inequality shows all of the numbers that satisfy the inequality. When graphing inequalities on a number
line, use solid circles ( ) for  and  and open circles ( ) for > and <.
Remember!

8. Additional Example 1A: Adding and Subtracting to Solve Inequalities
Course 3
2-5
Solving Inequalities Containing Integers
Additional Example 1A: Adding and Subtracting to Solve Inequalities
Solve and graph.
A. k +3 > –2 –3
k +3 > –2
Subtract 3 from both sides.
k > –5 –5

9. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Additional Example 1B: Adding and Subtracting to Solve Inequalities Continued
Solve and graph.
B. r – 9  12
r – 9  12
r – 
Add 9 to both sides.
r  21 15 21 24

10. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Additional Example 1C: Adding and Subtracting to Solve Inequalities Continued
Solve and graph.
C. u – 5  3
u – 5  3
u –  3 + 5
Add 5 to both sides.
u  8 5 8 10

11. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Additional Example 1D: Adding and Subtracting to Solve Inequalities Continued
Solve and graph.
D. c + 6  2
c + 6  2 –6
Subtract 6 from both sides.
c  –4 –7 –4 4

12. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Try This: Example 1A
Solve and graph.
A. y + 7  –1 –7
y + 7  –1
Subtract 7 from both sides.
y  –8
–11 –8

13. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Try This: Example 1B
Solve and graph.
B. f – 4  8
f – 4  8
f –  8 + 4
Add 4 to both sides.
r  12 12 15 6

14. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Try This: Example 1C
Solve and graph.
C. r – 2 < 1
r – 2 < 1
r – < 1 + 2
Add 2 to both sides.
r < 3 –3 3 8

15. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Try This: Example 1D
Solve and graph.
D. d + 9  6
d + 9  6 –9
Subtract 9 from both sides.
d  –3 –3 –7 4

16. 2-5 Solving Inequalities Containing Integers 5 > –1
Course 3
2-5
Solving Inequalities Containing Integers
Sometimes you must multiply or divide to isolate the variable. Multiplying or dividing both sides of an inequality by a negative number gives a surprising result.
5 > –1
5 is greater than –1.
–1 • 5 –1 • (–1)
Multiply both sides by –1. –
> or < ?
–5 < 1
You know –5 is less than 1, so you should use <.
–5 < 1
–7 –6 –5 –4 –3 –2 –
5 > –1

17. MULTIPLYING INEQUALITIES BY NEGATIVE INTEGERS
Course 3
2-5
Solving Inequalities Containing Integers
3 > 1
–4  12
Multiply by –2
Divide by –4
1  –3
–6 < –2
MULTIPLYING INEQUALITIES BY NEGATIVE INTEGERS
Words
Original Inequality
Multiply/Divide
Result
Multiplying or dividing by a negative number reverses the inequality symbol.

18. Course 3
2-5
Solving Inequalities Containing Integers
The direction of the inequality changes only if the number you are using to multiply or divide by is negative.
Helpful Hint

19. Additional Example 2: Multiplying and Dividing to Solve Inequalities
Course 3
2-5
Solving Inequalities Containing Integers
Additional Example 2: Multiplying and Dividing to Solve Inequalities
Solve and graph.
A. –3y  15
Divide each side by –3;  changes to .
–3y  15 –3 –7 –5 4
y  –5
B. 7m < 21
7m < 21 7
Divide each side by 7. 3 –3 5
m < 3

20. 2-5 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Try This: Example 2
Solve and graph.
A. –8y  24
Divide each side by –8; changes > to <.
–8y  24 –8 –3 –7 4
y  –3
B. 9f > 45
9f > 45 9
Divide each side by 9. 5 10
f > 5

21. 2-5 –2 –3 –3 Solving Inequalities Containing Integers Solve and graph.
Course 3
2-5
Solving Inequalities Containing Integers
Lesson Quiz: Part 1
Solve and graph.
1. h + 2 < 0 –2 2
h  –2
2. c – 5 > –2 3 –3
c  3 t –3
< 1 3 –3
t > –3 4.
7n > 28 4 8
n > 4

22. 2-5 5. x  2°F Solving Inequalities Containing Integers
Course 3
2-5
Solving Inequalities Containing Integers
Lesson Quiz: Part 2 5.
A local weather forecast stated that it would be 12°F tonight and at least 10° colder the next night. Write an inequality to show how cold it will be?
x  2°F