|x-3| - 5 >= 2
add 5 to both sides of the equation to get:
|x-3| >= 7
by definition, this means that:
(x-3) >= 7 and -(x-3) >= 7
(x-3) >= 7 becomes x-3 >= 7.
add 3 to both sides of that equation to get:
x >= 10
-(x-3) >= 7 becomes -x + 3 >= 7.
subtract 3 from both sides of that equation to get:
-x >= 4
multiply both sides of that equation by -1 to get:
x <= -4
that's because multiplying both sides of an inequality by a negative number reverses the inequality.
you get:
x >= 10 and x <= -4
when x = 10, |x-3| - 5 >= 2 becomes 2 >= 2 which is true.
when x = 12, |x-3| - 5 >= 2 becomes 4 >= 2 which is true.
when x = -4, |x-3| - 5 >= 2 becomes |-7| - 5 >= 2 becomes 7 - 5 >= 2 which becomes 2 >= 2 which is true.
when x = -6, |x-3| - 5>= 2 becomes |-9| - 5 >= 2 which becomes 9 - 5 >= 2 which becomes 4 >= 2 which is true.
when x is greater than -4 and less than 10, like x = 0, then:
|x-3| - 5 >= 2 becomes |-3| - 5 >= 2 which becomes 3 - 5 >= 2 which becomes -2 >= 2 which is NOT true.
the inequality is only true when x <= -4 and x >= 10.
this can be seen grephically below.
for your second problem, a similar procedure is taken.
start with |2x + 12| = -6
since the absolute value can never be negative, there is no solution to this problem.
if you tried to solve it, you would get the following.
this is true if and only if (2x + 12) = -6 and -(2x + 12) = -6
when (2x + 12) = -6, you get:
2x + 12 = -6 becomes 2x = -18 which becomes x = -9.
when x = -9, |2x + 12| = -6 becomes |-18 + 12| = -6 which becomes|-6| = -6 which becomes 6 = -6 which is NOT true.
when -(2x + 12) = -6, you get:
-2x - 12 = -6 which becomes -2x = 6 which becomes -x = 3 which becomes x = -3.
when x = -3, |2x + 12| = -6 which becomes |-6 + 12| = -6 which becomes |6| = -6 which becomes 6 = -6 which is NOT true.
there is no solution because the absolute value of an expression has to be greater than or equal to 0, which it is not when the absolute value is equal to -6.