When we find the absolute value of something, we are finding its "distance from zero." This is why both |3| = 3 and |-3| = 3, as more than just "turning them positive," we are stating that on a number line both 3 and -3 are three units from zero in either direction. If we solved an absolute value equation that said |x| = 3, we can create either condition for the scenario that x = 3 or x = -3 are possible. This will be true for any value |x| = c where c is positive. This is also why if |x| = 0 there is only one possible outcome since |0| = 0 is the only input that can yield that output, and this is why if c is negative we wouldn't have any solution to the equation (because taking the absolute value of something can never become negative).
In bigger absolute value equations though, such as -2|3x - 4| + 5 = -6, a couple of things will happen and the results will look a little different, especially on a number line. For one, once again, taking the absolute value of something doesn't just mean "turning it positive," so |3x - 4| ≠ 3x + 4 for example, as it is an entire quantity and not just a collection of different terms (plus, what if x was still a negative number?). Secondly, we cannot distribute the -2 into the absolute value as the function does indeed protect itself from potentially negative factors penetrating it. Lastly, we do not assume there is no solution unless we isolate the absolute value itself, and in that particular example you would get a positive right side of the equation if doing so, which would mean two solutions.
When solving, x won't be a value the same units left/right of zero, but it will be displaced from 0 since x is a part of the larger quantity (3x - 4) that is being absolute-valued. A lot of our equations will work on finding that middle value and how much we add/subtract from there as a whole, and many of the word problems involve us writing our own equations from information provided or graphed on a number line. In the future, not all absolute value questions will focus around this, but this is a great way of making the most sense of true "distance from a value" as possible.
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*TIMESTAMPS (separated by section)*
(0:00:00) Introduction
(0:01:08) Lecture overview
(0:16:39) Problem #1-2
(0:18:46) Problem #3-10
(0:23:26) Problem #11-24
(0:45:38) Problem #25-26
(0:51:10) Problem #27-30
(0:54:08) Problem #31-34
(0:58:12) Problem #35-44
(1:19:29) Problem #45
(1:22:35) Problem #46
(1:25:01) Problem #47-48
(1:30:26) Problem #49-50
(1:33:21) Problem #51
(1:36:33) Problem #52
(1:37:23) Problem #53-56
(1:42:10) Problem #57
(1:45:24) Problem #58
(1:47:04) Problem #59
(1:49:21) Problem #60
(1:51:49) Problem #61
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