From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them.
The absolute value function is a piecemeal function that does nothing to positive numbers, and inverts negative numbers, so it returns all positive numbers.
Absolute value can be expressed as a function: f(x) = |x| = x if x is positive, = -x if x is negative. The domain includes all real numbers (absolute value can be expanded to more numbers once we introduce them), and the range is all non-negative numbers (all positive numbers and 0).
This function is the same as asking how far a given number is from zero: both 6 and -6 are 6 units away from zero so |-6| = |6| = 6.
Absolute value can also be used to 'measure' the distance between two numbers: |a-b| would return how many units apart a and b are. Let's use 7 and 9. We intuitively know that 7 is two units away from 9 and 9 is two units away from 7. Here is how we see that with absolute values: |9-7| = |2| = 2 and |7-9| = |-2| = 2
Absolute value functions graphed exhibit a sharp switch from positive to negative, forming a 'V' shape.
This is the graph of y = |x|. Notice that the y-values remain positive when the x-values become negative.
Transformations of the absolute value graph
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A translation of the absolute value function is a shift in a cardinal direction. Subtraction of a from x (inside the absolute value) will shift the graph a units to the right. Subtraction of b from y (or addition outside the absolute value sign) will shift the graph b units up. The graph of y - b = |x - a| is the same as the shown one, except a units to the right and b units up.
A negative sign outside the absolute value sign will 'flip' the graph upside down, so the range of the function is all negative numbers instead of all positive.
Multiplying the whole function by c will make the "V" wider or narrower. If c is greater than 1, it will be narrower, if c is less than 1 it will be wider.
Solving Equations with Absolute Value
To algebraically solve an equation with absolute value, it should be solved with both halves of the piece-mail equation. Solve for the absolute value and replace with +/-. Remember that this will almost always create 'false' solutions, so solutions must be 'checked.'
- |3a - 5| - 6 = 3y - 2
- |3a - 5| = 3y + 4
- 3a - 5 = 3y + 4 or 3a - 5 = -(3y + 4)
From here, two different equations need to be solved:
- 3a - 5 = 3y + 4
- 3y = 3a - 9
- y = a - 3
And the second equation:
- 3a - 5 = -(3y + 4)
- -3y - 4 = 3a - 5
- -3y = 3a - 1
- y = -a + 1/3
We know how to calculate y, but it still depends on what a is.
If 3a - 5 if positive, y = a - 3
- 3a - 5 ≥ 0
- 3a ≥ 5
- a ≥ 5/3
For any value of a ≤ 5/3, y = -a + 1/3
The next example contains an inconsistent equation:
- |2k + 6| + 3 = 0
- |2k + 6| = -3
Here, there is no valid solution since the absolute value can only equal a positive number. It may appear that you can continue the same operations, but if you check the result, you will discover that the left and right sides will not match.