Question

Which set of terms best represents the graphed solution for the inequality |x−5|+1>5 ?

Option #1: open circles on the end values; two separate solution sets

Option #2: open circles on the end values; a single defined solution set

Option #3: closed circles on the end values; two separate solution sets

Option #4: closed circles on the end values; a single defined solution set

(1 point)

Answer 1

To solve the inequality \( |x - 5| + 1 > 5 \), we first isolate the absolute value:

\[
|x - 5| > 5 - 1
\]
\[
|x - 5| > 4
\]

This inequality indicates that the expression inside the absolute value, \( x - 5 \), is either greater than 4 or less than -4. Therefore, we can set up two separate inequalities:

1. \( x - 5 > 4 \)
2. \( x - 5 < -4 \)

Solving the first inequality:

\[
x - 5 > 4 \implies x > 9
\]

Solving the second inequality:

\[
x - 5 < -4 \implies x < 1
\]

So, the solution to the inequality \( |x - 5| + 1 > 5 \) is:

\[
x < 1 \quad \text{or} \quad x > 9
\]

Graphically, this means we have open circles at the points \( x = 1 \) and \( x = 9 \), and the solution set is represented by two separate intervals: \( (-\infty, 1) \) and \( (9, \infty) \).

Based on this analysis, the correct option is:

**Option #1: open circles on the end values; two separate solution sets**.