2-5 Solving Inequalities Containing Integers

Course 3

Lesson Presentation

Learn to solve inequalities with integers.

Course 3

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.

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.

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?

Course 3

2-5 Solving Inequalities Containing Integers

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.

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!

Course 3

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

Course 3

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.

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Solving inequalities is almost identical to solving equations :

Solving Inequalities

2 1 7x 2 7 1x

Example 1

2 8x 4x x is any value less than 4

Even Better News !

10 Apr 202310 Apr 2023

Solving inequalities is almost identical to solving equations :

Solving Inequalities

2(2 3) 9x x

4 6 9x x

Example 2

3 6 9x 3 15x

x is any value greater than or equal to 5

15 3x

5x

–5 1

Course 3

2-5 Solving Inequalities Containing Integers

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

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.

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

Course 3

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

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Solving Inequalities

The only one to watch out for is when you are dividing by a negative

Example

8 – 3m < 2

-3m < -6

m -6 -3

Subtract 8 from each side

Divide across by -3 and change the Sign

So m > 2

>

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Solving Inequalities

Example 2

5( x – 1 ) - 8x ≥ - 17

5x – 5 – 8x ≥ - 17

- 3x - 5 ≥ - 17

- 3x ≥ - 12

x -12 -3

≤ So x ≤ 4

Course 3

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.

Course 3

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