98.4k views
2 votes
What is the solution of |x – 6| ≥ 1? 5 < x < 7 x ≤ –7 or x ≥ –5 x ≤ 5 or x ≥ 7 –7 < x < –5

1 Answer

1 vote

Answer:

(c) x ≤ 5 or x ≥ 7

Explanation:

You want the solution to |x -6| ≥ 1.

Unfold

The absolute value relation represents two relations, one for the domain x < 6, and one for the domain x ≥ 6.

x < 6

In this domain, the inequality becomes ...

-1 ≥ x -6

5 ≥ x . . . . . . add 6

x ≤ 5 . . . . . . . put x on the left

x ≥ 6

In this domain, the inequality is ...

x -6 ≥ 1

x ≥ 7

The disjoint solution sets are x ≤ 5 or x ≥ 7.

__

Additional comment

For |x -a| ≤ b, we can "unfold" this to the compound inequality ...

-b ≤ (x -a) ≤ b

copying the inequality symbol to the left side, and writing the opposite of the constant there.

We can do the same thing with the inequality ...

|x -a| ≥ b

but it doesn't really make sense as a compound inequality.

Instead, we have to write it as ...

-b ≥ (x -a) or (x -a) ≥ b

in recognition of the fact that the solution spaces are disjoint.

What is the solution of |x – 6| ≥ 1? 5 < x < 7 x ≤ –7 or x ≥ –5 x ≤ 5 or x ≥ 7 –7 &lt-example-1
User Peacepassion
by
8.4k points

No related questions found