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

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

An absolute value is denoted by ││ and expresses the distance from zero points.

The absolute value of a positive number - will always be the number itself.** For example:** $│2│= 2$

Absolute value of a negative number: will always be the same number, but positive.** For example:** $│-3│=3$

Note that the absolute value of a number will always be a positive number since distance is always positive.

**For example:**

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

**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.**

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

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:** $│x+7│=12$

We will ask ourselves which absolute value expression will be 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:**

$x+7=12$

$x=5$

**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 it 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 it.

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:**

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

Test your knowledge

Question 1

\( \left|x-10\right|=0 \)

Question 2

\( \left|x+1\right|=5 \)

Question 3

\( \left|x-1\right|=6 \)

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