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

Question

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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)
Option #

Answer 1

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

\[ |x−5| + 1 > 5 \]

Subtracting 1 from both sides gives:

\[ |x−5| > 4 \]

Now, we can break this down into two separate inequalities:

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

Solving these inequalities:

1. For \(x - 5 > 4\): \[ x > 9 \]

2. For \(x - 5 < -4\): \[ x < 1 \]

Thus, the solution set is \(x < 1\) or \(x > 9\).

Now, we can summarize the graph of this solution:

• For \(x < 1\) (which is an open interval), we would have an open circle at 1.
• For \(x > 9\) (which is also an open interval), we would have an open circle at 9.
• Since the solutions don’t include the endpoints, we have open circles at both ends.

Based on this analysis, the correct representation is:

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