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

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories