Question

Is the absolute value of 4-x >14 a disjunction or a conjunction?

Answer 1

To determine whether the inequality \(|4 - x| > 14\) represents a disjunction or a conjunction, we first need to break it down.

The absolute value inequality \(|A| > B\) implies two separate inequalities:

1. \(A > B\)
2. \(A < -B\)

In this case, we have \(A = 4 - x\) and \(B = 14\). Therefore, we can express this as:

1. \(4 - x > 14\)
2. \(4 - x < -14\)

Now, let's solve each inequality separately:

1. For \(4 - x > 14\):
\[
-x > 14 - 4
\]
\[
-x > 10 \implies x < -10
\]

2. For \(4 - x < -14\):
\[
-x < -14 - 4
\]
\[
-x < -18 \implies x > 18
\]

The solutions are \(x < -10\) or \(x > 18\).

Since these two scenarios represent different conditions under which the original absolute value inequality holds true, the overall inequality is a disjunction (indicated by "or").

Thus, \(|4 - x| > 14\) represents a **disjunction**.