Absolute Value : The absolute value of a
number is the number of units it is from 0 on
the number line. (Distance from 0)
|x| = Absolute
Value sign
Ex) |2| , |-2| = 2
2 units
-2
-1
2 units
0
1
2
For any real number n:
If n≥0, then |n| = n
If n≤0, then |n| = -n
1. |7|
3. |0|
|7| = 7
2. |-12|
|0| = 0
-12 is less than 0, so
|12| equals –(-12)
|-12| = -(-12) or 12
The absolute value bars can be
grouping symbols.
Evaluate |4x-5|+ 4.1 if x=-3
|4x-5|+4.1 =
=
Absolute value
=
of -17 is 17
=
=
|4(-3)-5|+4.1
|-12-5|+4.1
|-17|+4.1
17+4.1
21.1
The value is 21.1
Replace x with -3
Simplify within
absolute value bars
first
<Rules for absolute value equations>
For N:
|x|=N
•If N is greater than 0 (N>0), then there are two solutions
•If N is equal to 0 (N=0), then there is one solution
•If N is less than 0 (N<0), then there is no solution
!
Before solving for solutions, you should only leave
absolute value at one side.
Ex) |x-14|+5=20
|x-14|=15
1. If N is greater than 0 (N>0), then there are two solutions
Ex)
|x-25|=17
Means x-25=17 or x-25=-17
X-25=17
X-25=-17
X-25+25=17+25
X-25+25=-17+25
X=8
X=42
Check
|x-25|=17
Check
|x-25|=17
|42-25|=17
|17|=17
|8-25|=17
|-17|=17
17=17
✔
17=17
✔
2. If N is equal to 0 (N=0), then there is one solution
Ex) |x+15|=0
X+15=0
X=-15
Ex2) |x-6|+4=4
X-6=0
X=6
0 isn’t negative nor positive, so
there is only one solution, 0.
Check
|x+15|=0
|-15+15|=0
|0|=0
0=0
✔
|x-6|=0
Check
|x-6|+4=4
|6-6|+4=4
|0|+4=4
0+4=4
4=4
✔
3. If N is less than 0 (N<0), then there is no solution
Ex) |2x+7|+5=0
|2x+7|=-5
The solution set that has no
members is called empty set (∅)
No solution; absolute
value of a number is
always positive or zero
For equations like |x+4|=2x-2, this rule might not work
For N:
•If N is greater than 0 (N>0), then there are two solutions
•If N is equal to 0 (N=0), then there is one solution
•If N is less than 0 (N<0), then there is no solution
Ex) |x-2|=2x-10
Means x-2=2x-10 or x-2=-(2x-10)
X-2=-(2x-10)
X-2=2x-10
X-2=-2x+10
x-2-x=2x-10-x
x-2+2x=-2x+10+2x
-2=x-10
8=x
Check
|x-2|=2x-10
|8-2|=2(8)-10
|6|=16-10
✔
6=6
3x-2=10
3x=12
X=4
Check
|x-2|=2x-10
|4-2|=2(4)-10
|2|=8-10
2≠-2
Q: For hydrogen to be a liquid, its temperature
must be within 2 degrees C of -257 degrees C.
It will give positive
number and
negative number
Let b=temperature of hydrogen to be a liquid, then |b-257|=2
Solve it like
normal absolute
value equations
that have two
solutions
|b-257|=2
b-257=2
b=259
b-257=-2
b=255
Range between
possible answers and
the number given.
The solutions are 259 and 255. Temperature of hydrogen
must be in between 255 degrees C and 259 degrees C
to be a liquid
Q: Hypothermia and hyperthermia are similar words but have
opposite meanings. Hypothermia is defined as a lowered body
temperature. Hyperthermia means an extremely high body
temperature. Both are potentially dangerous conditions and occur
when a person’s body temperature is more than 8 degrees above or
below the normal body temperature of 98.6 degrees F. At what
temperatures do these conditions begin to occur?
Let n=normal body temperature. Then |n-98.6|=8
|n-98.6|=8
n-98.6=8
n=106.6
n-98.6=-8
n=90.6
The solutions are 106.6 and 90.6. Hypothermia occurs when the
body temperature is below 90.6 degrees F. Hyperthermia occurs
when the body temperature is above 106.6 degrees F.
Q: A machine is to fill each of several boxes with 16 ounces
of sugar. After the boxes are filled, another machine weighs
the boxes. If the box is more than 0.2 ounces above or
below the desired weight, the box is rejected. At what
weights will the box be rejected?
Let x=desired weight, then |x-16|=0.2
|x-16|=0.2
x-16=0.2
x=16.2
x-16=-0.2
x=15.8
The solutions are 16.2 and 15.8. The box will be rejected
above 16.2 ounces and below 15.8 ounces.
Q: Foust Honda is having a contest to win a new Honda Civic. To
win a chance at the car, you must guess the number of keys in a
jar within 5 of the actual number. The people who are within this
range get to try a key in the ignition of the Civic. Suppose there
are 697 keys in the jar. What are the highest and lowest guesses
that will win a chance at the car?
Let x=number of keys within range of 5, then |x-697|=5
|x-697|=5
x-697=5
x=702
x-697=-5
x=692
The lowest guess is 692 and the highest guess is 702.
If people guess numbers between 692 and 702, then
they would win a chance at the car.
Q: A thermometer comes with a guarantee that the stated
temperature differs from the actual temperature by no more
than 1.5 degrees Fahrenheit. Write and solve an equation to
find the minimum and maximum actual temperatures when
the thermometer states the temperature is 87.4 degrees
Fahrenheit.
Let d=actual temperature ±1.5, then |d-87.4|=1.5
|d-87.4|=1.5
d-87.4=1.5
d=88.9
d-87.4=-1.5
d=85.9
The actual temperature is between 85.9
degrees F and 88.9 degrees F.