Answers
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High Hopes
Barry-
They're not equivalent.
(vertical bars) represents the absolute value of x. How it works is that it turns negative numbers positive but leaves 0 and positive numbers alone (hence it gets a number's distance from 0 on the number line).
![[x]](https://tex.z-dn.net/?f=%5Bx%5D)
(square brackets) usually represents the floor function, which returns the largest integer that is less than or equal to x. (The floor of x can also be written as 
--- it depends on what your textbook/source says).
To solve 
, you first transform it into the equivalent equation 
. Then by definition of absolute value, there are only two solutions for the first equation: x = 10 or x = -10.
[x] = 10 has infinitely many solutions. For example, the floor of 10 is 10, so ![[10] = 10](https://tex.z-dn.net/?f=%5B10%5D%20%3D%2010)
, thus a solution for the second equation is x = 10
The floor of 10.1 is 10, so ![[10.1] = 10](https://tex.z-dn.net/?f=%5B10.1%5D%20%3D%2010)
, thus another solution for the second equation is x = 10.1.
The two equations do not have the same solution set (as x = 10.1 does not solve |x| - 3 = 7 but solves [x] = 10), so they're not equivalent.
The answer is 2:1 because at first your ratio would be 18:9 but if you divide that by 9 (for simplifying purposes) than its 2:1
