Question

Which set of terms best represents the graphed solution for the inequality |x|+5≥9 ?

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 \( |x| + 5 \geq 9 \), we can start by isolating the absolute value:

\[
|x| + 5 \geq 9
\]

Subtract 5 from both sides:

\[
|x| \geq 4
\]

The absolute value inequality \( |x| \geq 4 \) means that \( x \) is either less than or equal to -4, or greater than or equal to 4. This can be expressed as:

\[
x \leq -4 \quad \text{or} \quad x \geq 4
\]

When we graph this solution on a number line, we use closed circles at -4 and 4 (because the inequality is "greater than or equal to"), indicating that these points are included in the solution. The graph consists of two separate parts: one extending to the left from -4 and the other extending to the right from 4.

Based on this reasoning, the correct representation of the graphed solution corresponds to:

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