Introduction to Absolute Value
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What is Absolute Value?
The absolute value of a number is its distance from zero.
Absolute value and distance are always represented as positive values.
Notice that both $|8|$ and $|-8|$ equal $8$.
Example Problems:
| 1. $|-5\tfrac{1}{3}| $ | 2. $|5| + |-2|$ | 3. $3(|-9|$ $-$ $|-7|)$ | ||
| $\color{blue}{= 5\tfrac{1}{3}}$ | $5$ $+$ $2$ | $=$ $3(5$ $-$ $7)$ | ||
| since absolute value | $\color{blue}{= 7}$ | $=$ $3(-2)$ | ||
| is always positive. | $\color{blue}{= -6}$ |
Solving Absolute Value equations?
Let’s look at the equation $|x| = 12$
We know that both $|12|$ and $|-12|$ equal $12$.
So there are 2 possible solutions $\color{blue}{x = 12}$ or $\color{blue}{x = -12}$
Let’s look at another problem.
| Original problem: | $|x + 6| = 22$ | |
| To remove the absolute value signs | ||
| we need to write the equation twice | $x + 6 = 22$ | |
| once with the answer being positive | $x = 16$ | |
| and once with the answer being negative | $x + 6 = -22$ | |
| $x = -28$ | ||
| So there are 2 possible solutions | $\color{blue}{x = 16}$ or $\color{blue}{x = -28}$ |
One last example
| Original problem: | $|2x-3| = 17$ | |
| To remove the absolute value signs | ||
| we need to write the equation twice | $2x – 3 = 17$ | |
| once with the answer being positive | $2x = 20$ | |
| once with the answer being positive | $x=10$ | |
| and once with the answer being negative | $2x-3 = -17$ | |
| $2x = -14$ | $x=-7$ | |
| So there are 2 possible solutions | $\color{blue}{x = 10}$ or $\color{blue}{x = -7}$ |
