September 20, 2011At the end of today, you will be able toSolve inequalities and compound inequalities

Warm-up: Solve for x1. │5x│ - 38 = -8

2. -6x + 8 = -14x – 28

HW 1.5: Pg. 37 #15-23odd, 33, Pg. 44 #8, 27

1.5 Inequalities

An inequality is a number sentence with more than one solution.

For example: x > -4“The solution of the inequality are all real #s

greater than -4.”Graph:

20-15 -5 5 15-20 -10 10-20 200

Two ways to express your answer:

Set Builder Notation

“The set of all numbers x such that x is greater than 9”

x x 4Interval Notation

The infinity symbols +∞ and -∞ are used to indicate that a set is unbounded in the positive or negative direction. To show that the endpoint is not included in the set, a parenthesis is used.

( 4,)

“ ( “ -- does not include the value“ [ “ -- includes the value

20-15 -5 5 15-20 -10 10-20 200

Set Builder to Interval Notation

For example:

x x 320-15 -5 5 15-20 -10 10-20 200

( ,3]

x x 3

( ,3)

x x 3

[3,)

x x 3

(3,)

1.5 Solving InequalitiesExample 1:

We solve equations like this:

add 8 + 8 +84x = -8

divide 44 4x = -2

4x – 8 = -16

Are the steps the same for this inequality?

4x – 8 ≥ -16 + 8 +84x ≥ -84 4

x ≥ -2

What would the graphs look like for both solutions?

Write the solution in set builder and interval notation.

Solving Inequality PracticeSolve and Graph. Write in set builder and

interval notation.1.

2. 3 > n + 6

3. -4x + 8 ≤ -24

5143

k

#1 rule for inequalities:

If you multiply or divide by a negative on both sides, switch the sign!

For #3, do you remember the “extra” step to correctly find the solutions?

Compound Inequalities

Compound Inequalities are two inequalities combined into one set of solutions.

For example: {x| x ≥ 4 or x < 0}Graph the solutions:

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What are some solutions of the set?

What are not solutions of the set?

Solving and graphing an “or” compound inequality

Example 2: Solve: x + 4 ≤ -2 or -12x < 6 Step 1: Solve as two separate inequalitiesStep 2: Graph both inequalities on one graph.

- 4 -4

10-8 -4 0 4 8-10 -6 2 10-2-10 6

x ≤ -6-12 -12

x > 21

or

What are some solutions?

What are not solutions?

{x | x ≤ -6 or x > }21

You try…

Solve and graph the inequality. 4. m – 7 ≥ -3 or -2m + 1 ≥ 11

5. 3a + 8 < 23 or

14a 6 7

HW 1.5: Pg. 37 #15-23odd, 33, Pg. 44 #8, 27

Example 3 Solve and graph: - 9 ≤ 4y – 3 ≤ 13

Solving and graphing an “and” compound inequality

Step 1: Solve as two separate inequalities

+ 3+ 3 + 3

- 6 ≤ 4y ≤ 164 4 4

≤ y ≤ 423

Step 2: Graph both inequalities on one graph.

10-8 -4 0 4 8-10 -6 2 10-2-10 6

{ x | }

Practice…

Solve each inequality and find the solutions set on a number line.

6. -14 ≤ 3x – 8 ≤ 16

7. 22 < 6w – 2 < 82

HW 1.5: Pg. 37 #19-23odd, 33, Pg. 44 #8, 9, 27, 28, 29