Absolute Value

Absolute value is denoted by || and represents the distance from zero.
The absolute value of a positive number - will always be the number itself.
For example: $│2│= 2$
The 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 positive since distance is always positive.

Absolute Value in an Equation with a Variable

If we have an unknown or an expression with an unknown inside absolute value, we ask ourselves which expression will give us the desired equation value, split into cases and find the unknown.
For example in the equation: $│x+7│=12$
We ask ourselves, which expression in absolute value will equal 12.
The answer will be 12 or -12. (both 12 in absolute value equals 12 and -12 in absolute value equals 12).
Therefore, we'll take the entire expression and split it into two cases:
First case:
$x+7=12$
Let's solve:
$x=5$

Second case:
$X+7=-12$
Let's solve:
$x=-19$

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

Detailed explanation

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Test yourself on absolute value!

Determine the absolute value of the following number:

$\left|18\right|=$

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Solving Absolute Value Inequalities and Their Interpretation

Understanding absolute value is crucial when solving inequalities involving absolute values. An absolute value inequality expresses the range of values that a variable can take, based on its distance from zero on the number line. For example, consider the inequality $∣x∣<5$ . This inequality states that the absolute value of $x$ is less than 5, meaning $x$ is within 5 units of zero. Therefore, $x$ can be any value between -5 and 5, not including -5 and 5 themselves. In interval notation, this is written as $-5 < x < 5$ . To solve absolute value inequalities, we often split them into two separate inequalities: one representing the positive scenario and the other the negative. This method allows us to find all possible values of the variable that satisfy the original inequality. Mastering this technique is essential for accurately solving and graphing solutions to absolute value inequalities.

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Test your knowledge

Question 1

$\left|-2\right|=$

Question 2

$\left|3\right|=$

Question 3

$\left|0.8\right|=$

Examples with solutions for Absolute value

Exercise #1

Determine the absolute value of the following number:

$\left|18\right|=$

Video Solution

Step-by-Step Solution

The "absolute value" can be viewed as the distance of a number from 0.
Therefore, the absolute value will not change the sign from negative to positive, it will always be positive.

Answer

$18$

Exercise #2

$\left|-2\right|=$

Video Solution

Step-by-Step Solution

When we have an exercise with these symbols || we understand that it refers to absolute value.

Absolute value does not relate to whether a number is positive or negative, but rather checks how far it is from zero.

In other words, 2 is 2 units away from zero, and -2 is also 2 units away from zero,

Therefore, absolute value essentially "zeroes out" the negativity of the number.

|-2| = 2

Answer

$2$

Exercise #3

$\left|3\right|=$

Video Solution

Answer

$3$

Exercise #4

$\left|0.8\right|=$

Video Solution

Answer

$0.8$

Exercise #5

$\left|x\right|=$

Video Solution

Answer

$x$

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