Solving Compound Inequalities

Solving Absolute Value Inequalities

3

Example 1

17or 63 xxThis is a compound inequality. It is already set up to start solving the separate equations.Since it has an “or” between the two, just put both graphs on the final graph and write your answer in interval notation.

33x 8x

5 8

3 7 7

Solving Absolute Value Inequalities

2

Example 2

42

xor 63 x

This is a compound inequality. It is already set up to start solving the separate equations.Since it has an “or” between the two, just put both graphs on the final graph and write your answer in interval notation.

3

2x 8x

0 8

3

)2()2(

Solving Absolute Value Inequalities

4

123 x

Example 3

082 and 663 xx This is an and problem. With and problems, we need to find out where the two graphs intersect. That will be the answer.Draw both, and see where they cross.

88

4x

0 4

6

33 2

682 x2

4x

Solving Absolute Value Inequalities

3)4,3[

1239 x

Example 4

93312 x This is another type of compound inequality. Whatever you do to get the x by itself in the middle, you have to do it to all “sides” of the inequality.Since it is written with two inequalities in one sentence, it is understood to have an “and” between them. Therefore, solve, and find the intersection.

3 3

43 x

0 4

3

3 3 3

Solving Absolute Value Inequalities

4628)4

14 and 123)3

532

xor 153)2

28or 952)1

x

xx

x

xx

Practice. Answers:

Solving Absolute Value Inequalities

0

Example 5

4or x 0 x

Write the inequality that fits the given graph.

2 4

Solving Absolute Value Inequalities

7

Example 6

57 x

Write the inequality that fits the given graph.

0 5

Homework:Page 317/ 9-23 odd, 27-31 odd, 38-41

Solving Absolute Value Equations

Solving Absolute Value Equations

53 x

Example 1

53 xThis problem has absolute value bars in it. Anytime you see absolute value bars in an equation, you need to split the problem into two different problems. The first equation is the exact as the original except just erase the absolute value bars. For the second equation, just change the sign of the other side.

53 x3 3 3

2x38x

OR

Solving Absolute Value Equations

1563 x

Example 2

1563 x1563 x

6 6 6

3

93

x

x

6

7

213

x

x

OR

Solving Absolute Value Equations

1312 x1312 x

4

Example 3

9412 x Always make sure that the absolute value bars are alone first, so add the four to both sides before you split it into two.

1312 x1 1

7x

1

6x

OR

4

1142 x

2 2122 x

2 2

Solving Absolute Value Equations

32 x32 x

3

Example 3.5

923 x You cannot distribute numbers into the absolute values. Since the negative three is being multiplied times the absolute value bars, to get rid of them, we need to divide both sides by the negative three.

32 x2 2

5x

2

1x

OR

3

2

Solving Absolute Value Equations

Example 4

34 xBecause the absolute values can never equal a negative, there is no work involved on this problem.

Homework:Page 325/ 7-19 odd, 20

Bellwork

84210)4

14 and 1132)3

133

xor 14)2

17or 933)1

x

xx

x

xx

Practice. Answers:

Solving Absolute Value Inequalities and Compound Inequalities

Solving Absolute Value Inequalities

or

7

Example 6

52 x Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces.When you write it the second time, not only do you change the sign, but you also turn the inequality around.To decide if you use “and” or “or”, remember GO to LA. Greater thanOrLess thanAnd

52 x2

3x 7x

0 3

52 x2 2 2

With “or”, just put both inequalities on the final

graph.

Solving Absolute Value Inequalities

and

4

Example 7

62 x Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces.To decide if you use “and” or “or”, remember GO to LA. Greater thanOrLess thanAndWhen you write it the second time, not only do you change the sign, but you also turn the inequality around.

62 x2

8x 4x

0 8

62 x2 2 2

With “and”, find where the two inequalities

intersect, and put that on the final graph.

Solving Absolute Value Inequalities

or

2

Example 8

842 x842 x

122 x

0 6

842 x

6x 2x42 x

Solving Absolute Value Inequalities

Example 9

845 x When there is a negative on the other side of an absolute value inequality, the answer is either “no solution” or “all real numbers”.Because the absolute value will always be positive, if it is a greater than, it will be “all real numbers”. If there is a less than sign with the negative on the outside, the answer is “no solution”.

273 xExample 10

Solving Absolute Value Inequalities

and

2

Example 11

1236 x1236 x

63 x

0 6

1236 x

2x 6x183 x

Solving Inequalities

3

2

Example 12

932 x

YOU DO NOT BREAK THIS INTO TWO PROBLEMS BECAUSE THERE ARE NO ABSOLUTE VALUE BARS!!!

362 x

23x2

3 4

Homework:Page 331/ 1-19 odd, Page 119/ 1-19 odd