Asked by Fiona
Find all x such that neither 2x nor -20x is in the interval (-infty,-1).
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Answered by
GPT-4o mini
To solve the problem, we need to find all \( x \) such that neither \( 2x \) nor \( -20x \) is in the interval \( (-\infty, -1) \).
### Step 1: Analyze the inequalities
1. For \( 2x \) not to be in the interval \( (-\infty, -1) \):
\[
2x \geq -1
\]
To isolate \( x \):
\[
x \geq -\frac{1}{2}
\]
2. For \( -20x \) not to be in the interval \( (-\infty, -1) \):
\[
-20x \geq -1
\]
This can be rewritten as:
\[
20x \leq 1
\]
Dividing by 20 gives:
\[
x \leq \frac{1}{20}
\]
### Step 2: Combine the inequalities
We now have two inequalities to consider:
1. \( x \geq -\frac{1}{2} \)
2. \( x \leq \frac{1}{20} \)
These can be combined to form the compound inequality:
\[
-\frac{1}{2} \leq x \leq \frac{1}{20}
\]
### Conclusion
The values of \( x \) that satisfy both conditions are:
\[
\boxed{[-\frac{1}{2}, \frac{1}{20}]}
\]
### Step 1: Analyze the inequalities
1. For \( 2x \) not to be in the interval \( (-\infty, -1) \):
\[
2x \geq -1
\]
To isolate \( x \):
\[
x \geq -\frac{1}{2}
\]
2. For \( -20x \) not to be in the interval \( (-\infty, -1) \):
\[
-20x \geq -1
\]
This can be rewritten as:
\[
20x \leq 1
\]
Dividing by 20 gives:
\[
x \leq \frac{1}{20}
\]
### Step 2: Combine the inequalities
We now have two inequalities to consider:
1. \( x \geq -\frac{1}{2} \)
2. \( x \leq \frac{1}{20} \)
These can be combined to form the compound inequality:
\[
-\frac{1}{2} \leq x \leq \frac{1}{20}
\]
### Conclusion
The values of \( x \) that satisfy both conditions are:
\[
\boxed{[-\frac{1}{2}, \frac{1}{20}]}
\]
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