Solving Inequalities

Solving Inequalities

3

2

Example 1

932 x

These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow most of the same rules.3

62 x2

3x2

3 4

Work the problem out as if it has an equals sign. Then when you get an answer, sketch the graph on the number line and write your answer in interval notation.

Solving Inequalities

x3

4

Example 2

8325 xxWhy is this different from Example 1? Because it has an x on both sides of the inequality. However, it doesn’t change how you work the problem.

x3822 x

2102 x2

5 6

2 25x

Solving Inequalities

9

5

Example 3

2139 xThis is the first example of a problem that is a little different than solving equations.

9123 x

4 3

3 34x

Anytime you multiply or divide by a negative, you have to turn the inequality symbol around.

Solving Inequalities

x3 x3

Example 4

2343 xx For this problem, since the variables subtracted off, it’s either “no solution” or “all real numbers”.It is “no solution” if the statement leftover is false.It is “all real numbers” if the statement is true.

24

Solving Inequalities

x2

1

Example 5

)4(2)2(4 xxIs this “no solution” or “all real numbers”?Neither. It’s only one of those if the variables go away.

x2882 x

802 x8

0 1

2 20x

8284 xx

Solving Rational Equations

x5

3067 x

247 x

Example 6

53

5

3

2

xx Remember, to multiply fractions

times whole numbers, divide the bottom into the whole number, then multiply what’s left.

( )

10

6 30 x2x2x2

6 6

7

24x

7

24

7

237

25

Solving Inequalities

]2

25,()4

)3

),7[)2

)3,)(1

Practice:

354

3

2)4

)23(4)14(3)3

15865)2

1024)1

xx

xx

xx

x

Answers:

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.

Solving Absolute Value Equations

x2 x2xx 92

Example 5

xx 92 Whenever there are variables inside the absolute value bars AND outside, you HAVE TO CHECK YOUR ANSWERS!!!

x2

x9

x2

x3

ORxx 92

x 91 1

x39 3 3

Solving Absolute Value Equations

918

9

39)3(2

Example 5

xx 92 Remember the rules for checking. 1) Always go back to the original problem.2) Do not cross the equals sign.

3) Use order of operations on each side.

96

3

Hurray! Both of them worked.

99)9(2

9 3

xx 92

Solving Absolute Value Equations

x xxx 212

Example 6

xx 212 Do I have to check my answers here?

x x

x4

ORxx 212

x 12 x312 3 3

Solving Absolute Value Equations

2424

2424

)4(2124

Example 6

xx 212

88

88 Since this gives you a true statement, 4 works. Therefore, x = 4.

)12(21212

xx 212

Solving Absolute Value Equations

x xxx 212

Example 6

xx 212 How do you communicate that?Mark off the answers that don’t work, and circle the ones that do.What is your answer if NEITHER answer checks?

x x

x4

ORxx 212

x 12 x312 3 3

Solving Absolute Value Equations

x x392 xx

Example 7

392 xx

x x

2x

OR392 xx

39 x 393 x

33

9 9

12x9 9

63 x

Solving Absolute Value Equations

15924

1515

329)2(2

Example 7

392 xx

594

55

Since neither one works, there are NO SOLUTIONS.

3129)12(2

1515 55

392 xx

Solving Absolute Value Inequalities and Compound Inequalities

Solving Absolute Value Inequalities

or

7),3()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)8,4(

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),6[]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]6,2[

Example 11

1236 x1236 x

63 x

0 6

1236 x

2x 6x183 x

Solving Inequalities

3

2

),3(

Example 12

932 x

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

362 x

23x2

3 4

Solving Absolute Value Inequalities

3),8[)3,(

Example 13

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

3)4,3[

1239 x

Example 14

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

26)3

15123)2

952)1

x

x

x

x

)1,7)[4

)3

]9,1)[2

),2()7,)(1

Practice. Answers:

Solving Absolute Value Inequalities

),2()2,()20

),25()45,()19

4)18

)17

)16

]5

6,2)[15

),7[]3,)(14

]4,3)[13

]2,6)[12

),1()9,)(11

x

Your turn. Do the worksheet on Inequalities. You may work together. This is for homework if you do not finish during class. Show your work!!!

)10

)9

)2,()8

]5,)(7

),2)[6

]7,)(5

]16,)(4

]3,)(3

)5,)(2

),2)(1

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