| n + 2 | ≥ 1 please explain it if you can thank you

Answer:

  (n ≤ -3) ∪ (-1 ≤ n)

Step-by-step explanation:

I like to "unfold" the absolute value expression by copying the right-side expression to the left side (with the same comparison symbol) and negating its value. I do it this way because I find it easier to work the problem "all at once".

  -1 ≥ n +2 ≥ 1

Obviously, -1 ≥ 1 is not true, which means the solution to this inequality will be disjoint sections of the number line. A compound inequality of this nature is generally interpreted to mean the AND of the two inequalities. So, technically, this is an incorrect step. I choose to overlook that, and consider the expression to represent the two inequalities ...

  -1 ≥ n +2 . . . OR

  n +2 ≥ 1

Subtracting 2 from the above compound inequality gives ...

  -3 ≥ n ≥ -1

So, the solution is ...

  (n ≤ -3) ∪ (-1 ≤ n)

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Further explanation

The inequality symbol negates its content if that content is negative. So, the expression ...

  |n+2| ≥ 1

means ...

  ±(n +2) ≥ 1

This resolves to two cases:

  n +2 ≥ 1

and

  -(n +2) ≥ 1

The latter case is equivalent to ...

  n +2 ≤ -1

which can also be written as ...

  -1 ≥ n +2

A more technically correct solution process would identify the two cases and work them separately.

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In the graph, the red shading (with the solid edge) shows the solution with respect to the numbers on the x-axis. If you were to graph this on a number line, you would put solid dots at -3 and -1, and shade the line to their left and right, respectively. The blue curve shows the absolute value, and the green line shows y=1, so you can see that the shaded areas correspond to the absolute value being greater than or equal to 1.