To determine which range includes the value of [tex]\(\sqrt{42}\)[/tex], we need to check where [tex]\(\sqrt{42}\)[/tex] falls among the given options.
1. Checking the first range: [tex]\( (6, 7) \)[/tex]
The range [tex]\( (6, 7) \)[/tex] means we are checking if [tex]\( 6 < \sqrt{42} < 7 \)[/tex].
2. Checking the second range: [tex]\( (41, 43) \)[/tex]
The range [tex]\( (41, 43) \)[/tex] means we are checking if [tex]\( 41 < \sqrt{42} < 43 \)[/tex].
3. Checking the third range: [tex]\( (7, 8) \)[/tex]
The range [tex]\( (7, 8) \)[/tex] means we are checking if [tex]\( 7 < \sqrt{42} < 8 \)[/tex].
4. Checking the fourth range: [tex]\( (36, 49) \)[/tex]
The range [tex]\( (36, 49) \)[/tex] means we are checking if [tex]\( 36 < \sqrt{42} < 49 \)[/tex].
By evaluating each range, we can see that [tex]\( \sqrt{42} \)[/tex] falls within the first range, [tex]\( (6, 7) \)[/tex], since [tex]\( 6 < \sqrt{42} < 7 \)[/tex].
For the other ranges:
- [tex]\( 41 < \sqrt{42} < 43 \)[/tex] is incorrect because [tex]\(\sqrt{42}\)[/tex] is much smaller than 41.
- [tex]\( 7 < \sqrt{42} < 8 \)[/tex] is incorrect because [tex]\(\sqrt{42}\)[/tex] is less than 7.
- [tex]\( 36 < \sqrt{42} < 49 \)[/tex] encompasses a much larger range, but as we're looking for the precise bounds that contain [tex]\(\sqrt{42}\)[/tex], this is too broad for our needs.
Therefore, only the first range makes the inequality true.
So, the values that make the inequality [tex]\(\,<\sqrt{42}<\)[/tex] true are:
[tex]\[ (6, 7) \][/tex]
