View the absolute value of a number as its distance from zero. When you take the absolute value of a number, you always end up with a positive number (or zero). Whether the input was positive or negative (or zero), the output is always positive (or zero). For instance, | 3 | = 3, and | _3 | = 3 also. This property that both the positive and the negative become positive makes solving absolute-value equations a little tricky. But once you learn the "trick", they're not so bad. Let's start with something simple: Solve | x | = 3 This is already solved: | 3 | = 3 and | _3 | = 3, so x must be 3 or _3. But how are you supposed to solve this if you don't already know the answer? You use the positive/negative property of the absolute value to split the equation into two cases, and you use the fact that the minus sign " _ " indicates "the opposite sign", not necessarily a negative number. For example, if you have x = _6, then " _x " indicates "the opposite of x", or, in this case, _(_6) = +6, a positive number. The minus sign in " _x " just indicates that you are changing the sign on x. It does not indicate a negative number. This distinction can be crucial. Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might have been positive and might have been negative, you have to acknowledge this fact when you take the absolute-value bars off, and you do this by splitting the equation into two cases. If the value of x was positive to start with, then you can bring that value out of the absolute-value bars without changing its sign, giving you x = 3. But x might also have been negative, in which case you would have to change the sign on x for the absolute value to come out positive, so you also have _x = 3, which solves as x = _3. Then the solution is x = _3, 3.
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