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8/7/2019 1-41-6_Abs_Value_Intro_Teacher [Autosaved]

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ABSOLUTE VALUEEQUALITIES and

INEQUALITIES

Candace Moraczewski and Greg Fisher

April, 2004


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3 x !

-3 3

This absolute value equation represents thenumbers on the number line whose distance from 0is equal to 3.

0

3 units 3 units

Two numbers satisfy this equation . Both 3 and -3 are3 units from 0 .

Look at the number line and notice the distance from 0

of -3

and3.

An absolute value equation is an equation that containsa variable inside the absolute value sign .


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The absolute value of a number is its distance

from 0 on a number line.

-5 0

5 5- ! because -5 is 5 units from 0

-3

3 3- ! because -3 is 3 units from 0


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0-5 -3

R emember though, on a number line if anumber is to the left of another number it isless than that number .

-5 < -3 even though 3- 5- "

R emember absolute value means distancefrom 0


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0 53

Notice that for numbers to the right of 0 thefollowing is true .

5 > 3 and also 3 5 "

R emember absolute value means distancefrom 0


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Solve | x +2 | = 7

x +2 = 7 or x+2 =-7-7 70

X+2

{5 ,-9 }x=

5 or x =

-9

Exactly 7units of -2


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Solve 4|x 3 | + 2 = 104| x 3 | = 8

| x 3 | = 2x 3 = 2 or x-3 = -2

x = 5 or x= 1{1,5 }

Exactly 2units of + 3


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Solve -2|2x + 1| -3 = 9

-2| 2x + 1| = 12

| 2x + 1| = -6

NO SOLUTION BecauseAbs . value cannot be negative

0


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-3 30

x

If a number x is between -3 and 3 then this translates to:

Inequality notation : -3 < x < 3 (a double inequality)

Absolute value notation : 3 x

because -3 is to the left of x and x is to the left of 3

because all of the numbers between -3 and 3 have adistance from 0 less than 3


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-3 30

x

If a number x is between -3 and 3 , including the -3 and 3 ,

then this translates to:

Inequality notation :-3

x3

(a double inequality)

Absolute value notation : 3 x e

e e


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-3 30

x

If a number x is to the left of -3 or to the right of 3 , includingthe -3 and 3 , then this translates to:

Inequality notation : x -3 or x 3 (a compound or inequality)

Absolute value notation : 3 x u

x

e u

- g


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2 xThis absolute value inequality represents all of thenumbers on a number line whose distance from 0 isless than 2 . See the red shaded line below .

0-2 2

Inequality notation : -2 < x < 2

x


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2 x e

0-2 2

This absolute value inequality represents all of thenumbers on the number line whose distance from 0is less than or equal to 2 . Notice that both -2 and 2are included on this interval .

Inequality notation : 2x2 ee

x


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2 x "

0-2 2

This absolute value inequality represents all of thenumbers on the number line whose distance from 0is more than 2 . Notice that the intervals satisfyingthis inequality are going in opposite directions .

Inequality notation : x < -2 or x > 2

x x

g- g


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2 x u

0-2 2

This absolute value inequality represents all of thenumbers on the number line whose distance from 0is more than or equal to 2 . Notice that the intervalssatisfying this inequality are going in oppositedirections and that 2 and -2 are included on theintervals .

Inequality notation : 2or x - 2x ue

xx

g- g


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W ord Problems Although I am endorsing this behavior,pretend that you are allowed to go within 9 of the speed limit of 55 mph without getting

a ticket.

Write an absolute value inequalitythat models this situation .

|x 65 | < 9

D esired amount Acceptable R ange

Check Answer: x -65 < 9 AND x-65 > -9

x 56 56


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W ord Problems

If a bag of chips is within .4 oz of 6 oz thenit is allowed to go on the market . W rite aninequality that models this situation .

|x 6 | < .4

D esired amount Acceptable R ange

Check Answer: x 6 < .4 AN D x 6 > -. 4

x < 6. 4 AN D x > 5.6 5.6 < x < 6. 4


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In a poll of 100 people, Mistys approvalrating as a dog is 78% with a 3 % of error . ticket . W rite an absolute value inequalitythat models this situation .

|x 78 | -3

x7 5 75


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TO SOLVE A MO R E COMPLICATE D ABSOLUTE VALUEINEQUALITY, FOLLO W THESE STEPS ASILLUST R ATE D IN THE FOLLO W ING EXAMPLES

1. D raw a number line and identify theinterval(s) which satisfy the inequality

2. W rite the expression in the absolute valuesign over the designated interval(s)3. Translate this to either a double inequality or two inequalities going in opposite directionsconnected with the word or

4. R emember to include the endpoint if theinequality also has an equal to symbol


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S olve 4 1 -x2

0-4 4

2x - 1

4 1-2 x4

Now solve the double inequality

1 . Draw a number line and identify the interval(s) which

satisfy the inequality :

2 . Write the expression in the absolute value signover the designated interval(s)

3. Translate this to either a double inequality or twoinequalities going in opposite directions


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S olve 8 2 3x e

0-8 8

3 x + 2

8 2 3x8 ee

Now solve the double inequality

3. Translate this to either a double inequality or two inequalities going in opposite directions

1 . Draw a number line and identify the interval(s)

which satisfy the inequality

2 .. Write the expression in the absolute value sign over the designated interval(s)


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8 2 3x8 ee

D ivide every position by 3

2 x310 ee

-2

-2

-2________________ 6 3x10 ee


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S olve 5 2 x u

0-5 5

x + 2

5 2 or x5- 2 x ue

Now solve the or compound inequality

x + 2


satisfy the inequality

2 . Write the expression in the absolute value signover the designated interval(s)

3. Translate this to either a double inequality or twoinequalities going in opposite directions

g- g


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5 2 or x5- 2 x ue-2 -2 -2 -2

3 or x7- x ue


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S olve 2 3x- 4 "

0-2 2

4 3 x

2 3x4or 2- 3x- 4 "

Now solve the or compound inequality

4 3 x


satisfy the inequality

2 . Write the expression in the absolute value signover the designated interval(s)3. Translate this to either a double inequality or twoinequalities going in opposite directions

g- g


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-4 -4 -4 -4

2- 3x-or 6- 3x- "

2 3x4or 2- 3x- 4 "

D ivide both inequalities by -3 . R emember to change

the sense of the inequality signs because of divisionby a negative .

3

2 or x2 x "


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TRY THE FOLLOWING P R OBLEM S. TO S EE THEANS WE R, CLICK ON THE A RR OW TO THE R IGHTOF THE P R OBLEM .

2 3x- 4 .6

1 1 2 x.51 2 x- 7- .4

5 3x- 2- .3

4 3 -x.2

5 3 - 2 .1

u

u

u

ex

Solve the following absolute value inequalities . W rite answer using both inequality notation and interval notation .

Click here to takethe post quiz


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ANS WE RS:

] 4 1,- [ , 4 x1- .1 ee

Click here to returnto the problem set


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) [7, 1]- ,- ( , 7 or x1- x2 . ggue

ANS WE RS:



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) 1 , 37- ( , 1 x

37- 3.

ANS WE RS: Click here to returnto the problem set


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) 3,- [ 4]- ,- ( , 3- or x4- x4. ggue

ANS WE RS:



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0) 1,- ( , 0 x1- 5 .

ANS WE RS:Click here to returnto the problem set


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) 2 , [ ] 32 ,- ( , 2 or x

32 x6. ggue

ANS WE RS:


1-41-6_abs_value_intro_teacher [autosaved]

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