To solve the inequality [tex]\( 3^{2x} \left( 81^x \right) \geq 27 \)[/tex], we need to follow a series of mathematical steps. Here's how we can proceed step-by-step:
1. Rewrite terms with the same base:
Notice that [tex]\( 81 \)[/tex] and [tex]\( 27 \)[/tex] are powers of 3:
- [tex]\( 81 = 3^4 \)[/tex]
- [tex]\( 27 = 3^3 \)[/tex]
So the inequality becomes:
[tex]\[
3^{2x} \left( 3^4 \right)^x \geq 3^3
\][/tex]
2. Simplify the exponents:
Use the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
3^{2x} \left( 3^{4x} \right) \geq 3^3
\][/tex]
Combine the exponents on the left side since the base is the same:
[tex]\[
3^{2x + 4x} \geq 3^3
\][/tex]
[tex]\[
3^{6x} \geq 3^3
\][/tex]
3. Solve the inequality for the exponents:
Since the bases are the same, we can compare the exponents directly:
[tex]\[
6x \geq 3
\][/tex]
Solve for [tex]\( x \)[/tex] by dividing both sides by 6:
[tex]\[
x \geq \frac{3}{6}
\][/tex]
[tex]\[
x \geq \frac{1}{2}
\][/tex]
4. Solution:
The solution to the inequality [tex]\( 3^{2x} \left( 81^x \right) \geq 27 \)[/tex] is [tex]\( x \geq \frac{1}{2} \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] must be at least [tex]\( \frac{1}{2} \)[/tex] for the inequality to hold true.
