Question

if the replacement set is the set of integers, find the solution set for the inequality: ( x - 9 < -15 )
a ( {-8, -7, -6, dots} )
b ( {-9, -8, -7, dots} )
c ( {dots, -8, -7, -6} )
d ( {dots, -9, -8, -7} )

Explanation:

Step1: Solve the inequality

To solve \( x - 9 < -15 \), we add 9 to both sides of the inequality.
\[

$$\begin{align*} x - 9 + 9&< -15 + 9\\ x&< -6 \end{align*}$$

\]

Step2: Analyze the solution set

The solution \( x < -6 \) means we are looking for all integers less than -6. Let's analyze each option:

• Option A: \(\{-8, -7, -6, \dots\}\) includes -6 and numbers greater than -6, which does not satisfy \( x < -6 \).
• Option B: \(\{-9, -8, -7, \dots\}\) includes -9, -8, -7 and numbers greater than -7 (like -6, -5, etc.), but we need numbers less than -6, so this is incorrect.
• Option C: \(\{\dots, -8, -7, -6\}\) includes -6, which is not less than -6, so this is incorrect.
• Option D: \(\{\dots, -9, -8, -7\}\) represents all integers less than -6 (since -9, -8, -7 are less than -6 and the "..." indicates numbers more negative, i.e., smaller than -9), so this satisfies \( x < -6 \).

Answer:

D. \(\{ \dots, -9, -8, -7\}\)