Respuesta :
They're not equivalent.
[tex]|x|[/tex] (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).
[tex][x][/tex] (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 [tex]\lfloor x \rfloor[/tex] --- it depends on what your textbook/source says).
To solve [tex]|x| - 3 = 7[/tex], you first transform it into the equivalent equation [tex]|x| = 10[/tex]. 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 [tex][10] = 10[/tex], thus a solution for the second equation is x = 10
The floor of 10.1 is 10, so [tex][10.1] = 10[/tex], 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.
