Pre-Algebra

2-5 Solving Inequalities Containing Integers

2-5 Solving Inequalities Containing Integers

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

2-5 Solving Inequalities Containing Integers

Warm UpCompare. Use >, <, or =.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

> >

=<

>

1. –3 –4 2. 6 –2

3. –8 –5 4. –5 –5

5. 3 –3 6. –8 –2 <

Pre-Algebra

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

Pre-Algebra

2-5 Solving Inequalities Containing Integers

Learn to solve inequalities with integers.

Pre-Algebra

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.

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.

Adding rock salt to the ice lowers the freezing point and helps to freeze the ice cream mixture.

Pre-Algebra

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 –17, the ice will remain frozen.

Pre-Algebra

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!

Pre-Algebra

2-5 Solving Inequalities Containing Integers

k > –5

A. k +3 > –2

Subtract 3 from both sides.–3 –3

k +3 > –2

Solve and graph.

Additional Example 1A: Adding and Subtracting to Solve Inequalities

–5 0

Pre-Algebra

2-5 Solving Inequalities Containing Integers

B. r – 9 12

r 21

Add 9 to both sides.r – 9 + 9 12 + 9r – 9 12

21 2415

Additional Example 1B: Adding and Subtracting to Solve Inequalities Continued

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

C. u – 5 3

u 8

Add 5 to both sides.u – 5 + 5 3 + 5u – 5 3

8 1050

Solve and graph.

Additional Example 1C: Adding and Subtracting to Solve Inequalities Continued

Pre-Algebra

2-5 Solving Inequalities Containing Integers

c –4

D. c + 6 2

Subtract 6 from both sides.–6 –6c + 6 2

0 4–7 –4

Solve and graph.

Additional Example 1D: Adding and Subtracting to Solve Inequalities Continued

Pre-Algebra

2-5 Solving Inequalities Containing Integers

y –8

A. y + 7 –1

Subtract 7 from both sides.–7 –7y + 7 –1

Try This: Example 1A

0–11 –8

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

B. f – 4 8

r 12

Add 4 to both sides.f – 4 + 4 8 + 4f – 4 8

12 15 6

Try This: Example 1B

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

C. r – 2 < 1

r < 3

Add 2 to both sides.r – 2 + 2 < 1 + 2r – 2 < 1

–3 0 3 8

Try This: Example 1C

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

d –3

D. d + 9 6

Subtract 9 from both sides.–9 –9d + 9 6

–3 0–7 4

Try This: Example 1D

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

–5 1

5 is greater than –1.

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 Multiply both sides by –1.–1 • 5 –1 • (–1) > or < ?

You know –5 is less than 1, so you should use <.

–5 < 1

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

–5 < 1

5 > –1

Pre-Algebra

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.

Pre-Algebra

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

Pre-Algebra

2-5 Solving Inequalities Containing Integers

A. –3y 15

y –5

Divide each side by –3; changes to .

B. 7m < 21

m < 3

Divide each side by 7.

Additional Example 2: Multiplying and Dividing to Solve Inequalities

Solve and graph.

0 3–3 5

–3y 15–3 –3

7m < 21 7 7

–7 –5 0 4

Pre-Algebra

2-5 Solving Inequalities Containing Integers

A. –8y 24

y –3

Divide each side by –8; changes > to <.

B. 9f > 45

f > 5

Divide each side by 9.

Try This: Example 2

–8y 24–8 –8

9f > 45 9 9

–3 0–7 4

0 5 10

Solve and graph.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

Lesson Quiz: Part 1Solve and graph.1. h + 2 < 0

2. c – 5 > –2

7n > 28

t

–33. < 1

h –2

n > 4

t > –3

c 3

–2 0 2

3–3 0

3–3 0

4 8 04.

Pre-Algebra

2-5 Solving Inequalities Containing Integers

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?

Lesson Quiz: Part 2

5.

x 2°F