To solve the inequality [tex]\( x^2 < 49 \)[/tex], we need to determine the range of values for [tex]\( x \)[/tex] that satisfy this condition. Let's go through the steps in detail:
1. Start with the given inequality:
[tex]\[
x^2 < 49
\][/tex]
2. Recall that if [tex]\( x^2 < a^2 \)[/tex], then the inequality can be rewritten as:
[tex]\[
-a < x < a
\][/tex]
Here, [tex]\( a \)[/tex] corresponds to [tex]\( \sqrt{49} \)[/tex]. Since [tex]\( \sqrt{49} = 7 \)[/tex], we substitute [tex]\( 7 \)[/tex] for [tex]\( a \)[/tex]:
3. Substitute [tex]\( a = 7 \)[/tex] into the inequality:
[tex]\[
-7 < x < 7
\][/tex]
4. This compound inequality represents all the values of [tex]\( x \)[/tex] that lie between [tex]\(-7\)[/tex] and [tex]\( 7 \)[/tex], exclusive.
5. Now we interpret the inequality [tex]\(-7 < x < 7\)[/tex]. This means [tex]\( x \)[/tex] must be greater than [tex]\(-7\)[/tex] and less than [tex]\( 7 \)[/tex], simultaneously.
6. From the given options, identify which one fits this interpretation:
- Option A: [tex]\( x < 7 \)[/tex] or [tex]\( x > -7 \)[/tex] - Incorrect; this means [tex]\( x \)[/tex] could be anywhere except between [tex]\(-7\)[/tex] and [tex]\( 7 \)[/tex].
- Option B: [tex]\( x > 7 \)[/tex] and [tex]\( x < -7 \)[/tex] - Incorrect; this is impossible because [tex]\( x \)[/tex] cannot simultaneously be greater than [tex]\( 7 \)[/tex] and less than [tex]\(-7\)[/tex].
- Option C: [tex]\( x < 7 \)[/tex] and [tex]\( x > -7 \)[/tex] - Correct; this correctly describes the condition [tex]\(-7 < x < 7\)[/tex].
- Option D: [tex]\( x > 7 \)[/tex] or [tex]\( x < -7 \)[/tex] - Incorrect; this means [tex]\( x \)[/tex] could be anywhere except between [tex]\(-7\)[/tex] and [tex]\( 7 \)[/tex].
Thus, the correct answer is:
C. [tex]\( x < 7 \)[/tex] and [tex]\( x > -7 \)[/tex]
