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

Determine the absolute value of the following number:

$\left|-25\right|=$

Question 2

Solve for the absolute value of the following integer:

$\left|34\right|=$

Question 3

What is the value of $\left| -3.5 \right|$ ?

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

Determine the absolute value of the following number:

$\left|-25\right|=$

Step-by-Step Solution

The absolute value of a number is the distance of the number from zero on a number line, without considering its direction. For the number $-25$ , the absolute value is $25$ because it is 25 units away from zero without considering the negative sign.

Answer

$25$

Exercise #3

Solve for the absolute value of the following integer:

$\left|34\right|=$

Step-by-Step Solution

The absolute value of a number is always non-negative because it represents the distance from zero. Therefore, the absolute value of $34$ is $34$ .

Answer

$34$

Exercise #4

What is the value of $\left| -3.5 \right|$ ?

Step-by-Step Solution

The absolute value of a number is the distance of the number from 0 on a number line, regardless of direction. Therefore, the absolute value of $-3.5$ is the same as moving 3.5 units away from 0, which results in $3.5$ . Hence, $\left| -3.5 \right| = 3.5$ .

Answer

$3.5$

Exercise #5

$\left|3\right|=$

Video Solution

Step-by-Step Solution

To solve this problem, we will determine the absolute value of the number 3:

Step 1: Recognize that the number given is 3, which is a positive number.
Step 2: According to the rules of absolute values, the absolute value of a positive number is the number itself.
Step 3: Therefore, $|3| = 3$ .

In conclusion, the absolute value of 3 is $\mathbf{3}$ .

Answer

$3$

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Absolute Value

Absolute Value

Absolute Value in an Equation with a Variable

Test yourself on absolute value!

Solving Absolute Value Inequalities and Their Interpretation

Examples with solutions for Absolute value

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

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Absolute value