Answer: [tex]3 \le x \le 5[/tex]
x is any real number between 3 and 5, including both endpoints
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Explanation:
If A < B, then 1/A > 1/B. Applying the reciprocal flips the inequality sign.
For example, if 2 < 3, then 1/2 > 1/3. It might help to look at the decimal representations
1/2 = 0.500
1/3 = 0.333
We see that 1/2 is larger.
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The inequality [tex]x < y < z[/tex] is the same as [tex]x < y \ \text{ and } \ y < z[/tex].
We've broken the single inequality into two smaller parts.
The given inequality breaks down into [tex]\frac{1}{6} \le \frac{1}{x+1}[/tex] and [tex]\frac{1}{x+1} \le \frac{1}{4}[/tex]
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Let's solve the first inequality for x
[tex]\frac{1}{6} \le \frac{1}{x+1}\\\\6 \ge x+1\\\\6-1 \ge x\\\\5 \ge x\\\\x \le 5[/tex]
Repeat for the other inequality as well
[tex]\frac{1}{x+1} \le \frac{1}{4}\\\\x+1 \ge 4\\\\x \ge 4-1\\\\x \ge 3\\\\[/tex]
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We found that [tex]x \ge 3 \ \text{ and } \ x \le 5[/tex]
This is the same as [tex]3 \le x \ \text{ and } \ x \le 5[/tex] which combines to [tex]3 \le x \le 5[/tex]
