Question

write the absolute value of x-5,ifx is less than 5 with clear solutions
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Answer 1

Absolute Value Symbol

To show we want the absolute value we put "|" marks either side (called "bars"), like these examples:

|−5| = 5 |7| = 7

vertical bar The "|" can be found just above the enter key on most keyboards.

More Formal

More formally we have:

Absolute Value

Which says the absolute value of x equals:

x when x is greater than zero

0 when x equals 0

−x when x is less than zero (this "flips" the number back to positive)

So when a number is positive or zero we leave it alone, when it is negative we change it to positive using −x.

Example: what is |−17| ?

Well, it is less than zero, so we need to calculate "−x":

− ( −17 ) = +17

(Because two minuses make a plus)

Useful Properties

Here are some properties of absolute values that can be useful:

|a| ≥ 0 always!

That makes sense ... |a| can never be less than zero.

|a| = √(a2)

Squaring a makes it positive or zero (for a as a Real Number). Then taking the square root will "undo" the squaring, but leave it positive or zero.

|a × b| = |a| × |b|

Means these are the same:

the absolute value of (a times b), and

(the absolute value of a) times (the absolute value of b)

Which can also be useful when solving

|u| = a is the same as u = ±a and vice versa

Which is often the key to solving most absolute value questions.

Example: Solve |x+2| = 5

Using "|u| = a is the same as u = ±a":

this: |x+2| = 5

is the same as this: x+2 = ±5

Which has two solutions:

x+2 = −5 x+2 = +5

x = −7 x = 3