Absolute value
The absolute value of x describes the distance of x from zero. The absolute value is denoted by vertical lines, i.e. number . Figure 1 shows the absolute value of the number 3 drawn in red. When moving from zero to right, the number 3 is at the third "post". So the distance is 3 units and the absolute value of the number 3 is 3 = 3.
Blue represents the absolute value of -5. When moving from zero to the left, the number -5 is at the fifth "post". The distance is thus 5 units and the absolute value of -5 is -5 = 5.
The mathematical formula for absolute value is
Since the absolute value is distance, its value is always at least 0. The formula x embraces all that is written within absolute values. If it is an expression, then it must be put in parentheses (see example 3).
Examples:
1.
2. , as
3. as
Example 4.
Example 5. Simplify if
There two terms in this expression. Both terms are within absolute value. The value of x is between -5 and 5. Thus, the value of is always at most zero, because x is at maximum 5. For example, for a variable x with a value of 4, the value of the expression is 4 - 5 = -1.
Since the values are never positive, then we need to use the lower row of the formula. Since there is an expression within the absolute value, we put it in parentheses and a minus sign in front of it:
The second term is always at least zero because the value of x is at least -5. For example -3 + 5 = 2. So we can use the upper line of the formula:
Next, we will add these terms:
A reduced answer of 10 is significantly easier than the original expression. According to this, the value of the expression is always 10 when To check the answer, place the number 3 in the original expression:
Let's do the same with -4: