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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
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
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
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
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:
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