Absolute Value Formula

Absolute value formulas: It's essential to review the concept of absolute value. Absolute value measures a number's distance from zero on a number line. This measurement always results in a positive number or zero; it's never negative. We use the symbol |x| to represent the absolute value of a number "x," and Sometimes, you may come across the terms "mod x" or "modulus of x" to describe the same concept. So, when we find the absolute value of a number, we are finding its distance from zero, and that distance is always a positive value.

What is the Absolute Value Formula?

The absolute value formula helps us find the absolute value of any number, ensuring that the result is always positive. This formula is best explained through an example:

In the example above, when we calculate |4|, it equals 4. Therefore, we can conclude that |x| = x for all x values equal to or greater than 0.

Similarly, when we calculate |-4|, it equals 4. As a result, we can conclude that |-4| equals -(-4), which simplifies to 4. Therefore, we can state:

|x| = -x for every x less than 0.

|x| = x for every x greater than or equal to 0.

This is the formula for Absolute Value.'

Absolute Value Formula Examples

Example 1: Finding Absolute Values

Let's apply the absolute value formula to find the absolute values of -1/3, 5, and -0.5:

|-1/3| = 1/3

|5| = 5

|-0.5| = 0.5

So, the absolute values are |-1/3| = 1/3, |5| = 5, and |-0.5| = 0.5.

Example 2: Finding Solutions for the Equation |x| = 5

Now, let's solve the equation |x| = 5 using the absolute value formula:

Given equation: |x| = 5

According to the absolute value formula, |x| can be either +x or -x, depending on the sign of x. Therefore, we have:

+x = 5

-x = 5

Solving these equations separately:

For +x = 5, x = 5

For -x = 5, x = -5

The solutions to the equation |x| = 5 are x = 5 and x = -5.

Example 3: Find the absolute values of the following numbers: -2, 0, and 3.

Using the absolute value formula:

|-2| = 2

|0| = 0

|3| = 3

So, the absolute values are |-2| = 2, |0| = 0, and |3| = 3.

Example 4: Solving the Equation |x| = 10 Using the Absolute Value Formula

Given equation: |x| = 10

According to the absolute value formula:

+x = 10

-x = 10

Solving these equations separately:

For +x = 10, x = 10

For -x = 10, x = -10

The solutions to the equation |x| = 10 are x = 10 and x = -10.

Example 5: Calculate the absolute value of the difference between two numbers, a = 7 and b = 15.

Using the absolute value formula to find |a - b|:

|7 - 15| = |-8| = 8

So, the absolute value of the difference between 7 and 15 is 8.

Example 6: Determine the absolute value of a negative fraction, -1/4.

Using the absolute value formula:

|-1/4| = 1/4

Now, the absolute value of -1/4 is 1/4.

Example 7: Solve the absolute value equation |2x - 1| = 7 using the absolute value formula.

Given equation: |2x - 1| = 7

According to the absolute value formula:

2x - 1 = 7

-(2x - 1) = 7

Solving these equations separately:

For 2x - 1 = 7, 2x = 8, x = 4

For -(2x - 1) = 7, -2x + 1 = 7, -2x = 6, x = -3

Now, the solutions to the equation |2x - 1| = 7 are x = 4 and x = -3.

Example 8: Find the solutions to the equation |3x - 2| = 8.

Solution:

We will consider two cases:

Case 1: 3x - 2 is positive or zero:

3x - 2 = 8

Now, solve for x:

3x = 8 + 2

3x = 10

x = 10/3

Case 2: 3x - 2 is negative:

-(3x - 2) = 8

Now, solve for x:

-3x + 2 = 8

-3x = 8 - 2

-3x = 6

x = 6/(-3)

x = -2

So, the solutions to the equation |3x - 2| = 8 are x = 10/3 and x = -2.