Absolute Value

πŸ†Practice equation with absolute value

The "absolute value" may seem complicated to us, but it is simply the distance between a given number and the figure 0 0 .Β 

What is absolute value?

An absolute value is denoted by β”‚β”‚ and expresses the distance from zero points.
The absolute value of a positive number - will always be the number itself.
For example: β”‚2β”‚=2β”‚2β”‚= 2
Absolute value of a negative number: will always be the same number, but positive.
For example: β”‚βˆ’3β”‚=3β”‚-3β”‚=3
Note that the absolute value of a number will always be a positive number since distance is always positive.

The absolute value of a number is the distance between it and the number 0.

For example:

  • The distance between the number +7 +7 and 0 0 is 7 7 units. Therefore, the absolute value of +7 +7 is 7 7 .
  • The distance between the number βˆ’7 -7 and 0 0 is also 7 7 units. Therefore, the absolute value of βˆ’7 -7 will also be 7 7 .Β 

As we can see, from the point of view of absolute value, it doesn't matter if the number is positive or negative.

To denote the absolute value, the number is written between two vertical lines.

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Test yourself on equation with absolute value !

einstein

\( \left|6x-12\right|=6 \)

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Absolute value exercise

If we have an unknown or an expression with an unknown within an absolute value, we will ask ourselves which expression will bring us the value of the desired equation, we will divide into cases and discover the unknown.
Example in the equation: Β β”‚x+7β”‚=12β”‚x+7β”‚=12
We will ask ourselves which absolute value expression will be equal to 1212 .
The answer will be 1212 or βˆ’12-12 . (Both an absolute value is equal to 1212 and an absolute 1212 is equal to βˆ’12-12 ).
Therefore, we will take the complete expression and divide it into two cases:
First case:

x+7=12x+7=12
We solve:
x=5x=5

Second case:
X+7=βˆ’12X+7=-12
We solve
x=βˆ’19x=-19

Therefore, the solution to the exercise is: x=5,βˆ’19x=5,-19


Examples:

  • the absolute value of βˆ’7 -7 is represented as follows: ∣7∣ |7| ;
  • the absolute value of +9 +9 is represented as follows: ∣9∣ |9| .Β 

However, when writing calculations, we will do it as follows:

  • βˆ£βˆ’20∣=20|-20|= 20
  • ∣+13.6∣=13.6|+13.6|=13.6
  • ∣(+44)+(βˆ’5)∣=∣+39∣=39|(+44)+(-5)|=|+39|=39
  • βˆ£βˆ’9+6∣=βˆ£βˆ’3∣=3|-9+6|=|-3|=3
  • βˆ£βˆ’9∣+6=9+6=15|-9|+6=9+6=15
  • ∣(βˆ’56)∣+(βˆ’13)=56+(βˆ’13)=43|(-56)|+(-13)=56+(-13)=43
  • 28+∣4βˆ’9∣=28+βˆ£βˆ’5∣=28+5=3328+|4-9|=28+|-5|=28+5=33

The absolute value of a negative number will always be greater than it.Β 

The absolute value of a positive number will always be equal to the positive number.Β 

Examples:

  • βˆ£βˆ’6∣>βˆ’6|-6|>-6
  • ∣+6∣=+6|+6|=+6

Practice Exercises to Find the Absolute Value

Fill in the blanks with one of the following symbols:Β <, >, =.Β 

  • βˆ’4-4,Β β–―Β ~β–―~ ,βˆ£βˆ’4∣ |-4|
  • βˆ£βˆ’9∣|-9|,Β β–―Β ~β–―~ ,+6+6
  • βˆ£βˆ’50∣|-50|,Β β–―Β ~β–―~ ,∣+50∣|+50|
  • 8+58+5,Β β–―Β ~β–―~ ,βˆ£βˆ’14∣ |-14|
  • βˆ£βˆ’5βˆ’5∣|-5-5|,Β β–―Β ~β–―~ ,4+5 4+5
  • ∣+53∣|+53|,Β β–―Β ~β–―~ ,5353
  • βˆ£βˆ’3βˆ’2∣|-3-2| ,Β β–―Β ~β–―~ ,∣6βˆ’1∣|6-1|
  • ∣14+(βˆ’8)∣|14+(-8)|,Β β–―Β ~β–―~ ,14+βˆ£βˆ’8∣ 14+|-8|
  • 14+(βˆ’8)14+(-8),Β β–―Β ~β–―~ ,∣+14∣+(βˆ’8)|+14|+(-8)

Solve the following exercises:

  • βˆ£βˆ’5+6∣=|-5+6|=
  • 22βˆ’βˆ£βˆ’53∣=22-|-53|=
  • ∣15βˆ’19∣+βˆ£βˆ’9βˆ’7∣=|15-19|+|-9-7|=
  • ∣15βˆ’19βˆ’9βˆ£βˆ’7=|15-19-9|-7=
  • 15βˆ’βˆ£19βˆ’9βˆ’7∣=15-|19-9-7|=
  • 9.7βˆ’βˆ£4.3+(βˆ’6)∣=9.7-|4.3+(-6)|=
  • 9.7βˆ’4.3+βˆ£βˆ’6∣=9.7-4.3+|-6|=


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