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20 votes
What is the solution to |x − 4| ≥ 7? PLEASE HELP ME I NEED TO GRADUATE

A.
-3 ≤ x ≤ 11
B.
-11 ≤ x ≤ 3
C.
x ≥ 11 or x ≤ -3
D.
x ≥ 3 or x ≤ -11

User Darkglow
by
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1 Answer

15 votes
15 votes

Answer:

C. x ≥ 11 or x ≤ -3

Explanation:

This inequality can be resolved into two:

-7 ≥ x -4 or x -4 ≥ 7

Adding 4 to the first one of these gives ...

-3 ≥ x ⇔ x ≤ -3

Adding 4 to the second of the inequalities above gives ...

x ≥ 11

The full solution is then ...

x ≥ 11 or x ≤ -3 . . . . . matches choice C

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Additional comment

You can see in the attachment that when the value of the absolute value expression is supposed to be greater than some number, the solution set will have two disjoint parts. That means the answer expression must include "or", eliminating choices A and B.

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When you are given the inequality in the form ...

|f(x)| ≤ c . . . . . note "less than ..."

It can be resolved to the compound inequality ...

-c ≤ f(x) ≤ c

When the inequality symbol points the other way ("greater than ..."), this sort of resolution doesn't really make sense. (You can do it, but you need to recognize the nonsense involved. Your teacher may not appreciate this "work".) In the present case, that would look like ...

-7 ≥ x -4 ≥ 7

When you add 4 to all parts of this, it becomes ...

-3 ≥ x ≥ 11

Recognizing the nonsense in this expression, you can rewrite it in two parts the way it should be written:

-3 ≥ x OR x ≥ 11

What is the solution to |x − 4| ≥ 7? PLEASE HELP ME I NEED TO GRADUATE A. -3 ≤ x ≤ 11 B-example-1
User Parn
by
2.6k points