Find all x such that neither 2x nor -20x is in the interval (-infty,-1).

Question

Show drafts

Answer 1

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:

\[ \boxed{[-\frac{1}{2}, \frac{1}{20}]} \]