More on Absolute Values
When you work with absolute values, recall that the absolute value of a number is its distance from zero. Given the use of the number line, any number and its additive inverse possess the same absolute value. You represent this situation as follows:
|3| = 3
| – 3| = 3
When you translate this set of relations so that you explicitly identify the values of x in a formal way, you can create an expression that assumes this form:
If |x| = 3 ⇒ x = 3, – 3.
Such a statement reads, “Given that the absolute value of some number x is 3, then x can be either 3 and –3.” As Figure 5.4 illustrates, if you employ a number line, you can easily show the two values.
Figure 5.4. You can associate negative and positive numbers with a statement involving an absolute value.
With the use of a number line, you distinguish the absolute value with respect to the origin of the number line, which is zero. The absolute value of a number is its distance from the origin.
Given such an understanding, when you solve an equation that involves an absolute value, then you deal with two absolute distances. Consider this equation:
|x – 2| = 5
Working with this problem involves two solutions. You arrive at these solutions by anticipating that |x – 2| can be equal to either a positive or negative number. The positive or negative number in this instance is 5 or –5. Given this situation, your solution involves two equations:
| x – 2 = 5 | x – 2 = –5 |
| x – 2 + 2 = 5 + 2 | x – 2 + 2 = –5 + 2 |
| x = 7 | x = –3 |
To verify the correctness of these solutions, you substitute them back into the original equation:
|7 – 2| = |5| = 5 | – 3 – 2| = | – 5| = 5
Here is a second example. It works the same way. You begin with an equation that includes an expression that embodies an absolute value. You then solve the equation for the two values the absolute value allows:
|x + 7| = 8
You then proceed with solutions as follows:
| x + 7 = 8 | x + 7 = –8 |
| x + 7 – 7 = 8 – 7 | x + 7 – 7 = –8 – 7 |
| x = 1 | x = –15 |
As with the previous example, to test the correctness of these solutions, you substitute them back into the original equation:
|1 + 7| = |8| = 8 |– 15 + 7| = |– 8| = 8
Exercise Set 5.5 |
