Let's solve the inequality [tex]\(\left( \frac{1}{64} \right)^x + 15 < 271\)[/tex] for [tex]\(x\)[/tex].
1. Start by isolating the exponential term:
[tex]\[
\left( \frac{1}{64} \right)^x + 15 < 271
\][/tex]
Subtract 15 from both sides:
[tex]\[
\left( \frac{1}{64} \right)^x < 256
\][/tex]
2. Knowing that [tex]\(\frac{1}{64} = 64^{-1}\)[/tex], we can rewrite the inequality:
[tex]\[
64^{-x} < 256
\][/tex]
3. Express 256 as a power of 64. Since [tex]\( 256 = 4^4 \)[/tex] and [tex]\( 64 = 4^3 \)[/tex], we can write:
[tex]\[
256 = 4^4 = (4^3)^{4/3} = 64^{4/3}
\][/tex]
4. Substitute this back into the inequality:
[tex]\[
64^{-x} < 64^{4/3}
\][/tex]
5. Since the bases are the same, we can set the exponents to satisfy the inequality:
[tex]\[
-x < \frac{4}{3}
\][/tex]
6. Multiply both sides by -1 and reverse the inequality sign:
[tex]\[
x > -\frac{4}{3}
\][/tex]
Thus, the solution to the inequality [tex]\(\left( \frac{1}{64} \right)^x + 15 < 271\)[/tex] is:
[tex]\[
\boxed{x > -\frac{4}{3}}
\][/tex]
