Algebra: absolute value solving?

The vertical slash | is on the keyboard, usually on the shifted backslash \ key

3 - | (2x + 5)/3 | = 5

- | (2x + 5) - 3 | = 5 - 3

- | (2x + 5) - 3 | = 2

Now that is an extremely odd expression, to which there is no possible solution.

| (2x + 5) - 3 | has a positive value at all times. (That is the function of the "abs" sign | . | )

But there is a minus sign in front of it, which means that the left-hand side is always negative.

And yet the right-hand side of the equation is +2 !

There are no circumstances in which a negative number can equal a positive number, therefore there is something wrong with the original equation.

[ In fact, that can also be seen from the original expression, but I thought the above might make it a bit clearer.

If we let A = | (2x + 5)/3 |

then

3 - | (2x + 5)/3 | = 5 can be written as

3 - A = 5

Since A is an absolute value, it is always positive. And there is no way of subtracting a POSITIVE number from 3 to get 5 ! ]

2 - 4 |x + 2| ≥ -6

- 4 |x + 2| ≥ -8

|x + 2| ≤ -8/-4

|x + 2| ≤ 2

That expression means that

(x + 2) ≤ 2, OR that -(x + 2) ≤ 2

Treat each part separately.

(x + 2) ≤ 2 . . . . . . . . . . or . . . . . . . . . . -(x + 2) ≤ 2

x ≤ 4. . . . . . . . . . . . . . . . . . . . . . . . . . . .(x + 2) ≥ - 2 ( a (-1) reverses the ≤ sign)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x ≥ 0

Hence the solution set is 0 ≤ x ≤ 4

amounts not written properly