Absolute Value

If we have an unknown or an expression with an unknown within an absolute value, we should query which expression will result in the value of the desired equation. We will proceed to divide the problem into cases in order to discover the unknown.
Example in the equation:  $│x+7│=12$

We will check which absolute value expression is equal to $12$.
The answer will be $12$ or $-12$. (Both an absolute value is equal to $12$ and an absolute $12$ is equal to $-12$).
Therefore, we will take the complete expression and divide it into two cases:
First case:
$x+7=12$
We solve as follows:
$x=5$

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

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

Examples:

• The absolute value of $-7$ is represented as follows: $|7|$;
• The absolute value of $+9$ is represented as follows: $|9|$. 

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

• $|-20|= 20$
• $|+13.6|=13.6$
• $|(+44)+(-5)|=|+39|=39$
• $|-9+6|=|-3|=3$
• $|-9|+6=9+6=15$
• $|(-56)|+(-13)=56+(-13)=43$
• $28+|4-9|=28+|-5|=28+5=33$

The absolute value of a negative number will always be greater than the number itself. 

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

Examples:

• $|-6|>-6$
• $|+6|=+6$

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

• $-4$,$~▯~$,$|-4|$
• $|-9|$,$~▯~$,$+6$
• $|-50|$,$~▯~$,$|+50|$
• $8+5$,$~▯~$,$|-14|$
• $|-5-5|$,$~▯~$,$4+5$
• $|+53|$,$~▯~$,$53$
• $|-3-2|$,$~▯~$,$|6-1|$
• $|14+(-8)|$,$~▯~$,$14+|-8|$
• $14+(-8)$,$~▯~$,$|+14|+(-8)$

Solve the following exercises:

• $|-5+6|=$
• $22-|-53|=$
• $|15-19|+|-9-7|=$
• $|15-19-9|-7=$
• $15-|19-9-7|=$
• $9.7-|4.3+(-6)|=$
• $9.7-4.3+|-6|=$