If the replacement set is the set of integers, find the solution set for the inequality:
x−9<−15
The Correct Answer and Explanation is:
To solve the inequality ( x – 9 < -15 ) for ( x ) with the replacement set being the set of integers, we can follow these steps:
- Isolate ( x ):
We want to isolate ( x ) on one side of the inequality. To do this, add 9 to both sides:
[
x – 9 + 9 < -15 + 9
]
Simplifying this gives:
[
x < -6
] - Identify the Solution Set with Integers:
Now, we have the inequality ( x < -6 ). Since the replacement set is the set of integers, we are only interested in integer values of ( x ) that satisfy this inequality. The integer values less than -6 are:
[
-7, -8, -9, -10, \ldots
] Therefore, the solution set is:
[
{ x \in \mathbb{Z} \,|\, x < -6 } = {-7, -8, -9, -10, \ldots}
]
Explanation of Solution
This problem involves solving a basic linear inequality and finding the integer values that satisfy it. An inequality, unlike an equation, represents a range of possible solutions rather than a single solution. The goal here is to isolate the variable ( x ) on one side, allowing us to interpret the range of values it can take.
Starting with ( x – 9 < -15 ), we add 9 to both sides to isolate ( x ). Adding or subtracting the same value on both sides of an inequality preserves the inequality’s direction. After this step, we get ( x < -6 ), which tells us that ( x ) must be any value smaller than -6.
Since we are limited to the set of integers, we ignore any decimal or fractional values. The integers less than -6 begin at -7 and continue indefinitely in the negative direction, so our solution set includes all integers less than -6, which we can write as ( {-7, -8, -9, -10, \ldots} ).
This type of problem is foundational in algebra and helps build skills for understanding inequalities and solution sets, especially when restrictions are given, such as working within the integers.