To determine the solution set for the exponential inequality [tex]\(3^{2x - 1} < 27\)[/tex], follow these steps:
1. Rewrite the inequality involving the same base:
Recognize that 27 can be expressed as a power of 3:
[tex]\[
27 = 3^3
\][/tex]
So, the inequality becomes:
[tex]\[
3^{2x - 1} < 3^3
\][/tex]
2. Compare the exponents:
Since the bases are the same (both are base 3), you can compare the exponents directly to solve for [tex]\( x \)[/tex]:
[tex]\[
2x - 1 < 3
\][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
Begin by isolating [tex]\( x \)[/tex]:
[tex]\[
2x - 1 < 3
\][/tex]
Add 1 to both sides:
[tex]\[
2x < 4
\][/tex]
Divide both sides by 2:
[tex]\[
x < 2
\][/tex]
4. Interpret the solution:
The solution to the inequality [tex]\( 3^{2x - 1} < 27 \)[/tex] is all values of [tex]\( x \)[/tex] that are less than 2. In interval notation, this is expressed as:
[tex]\[
(-\infty, 2)
\][/tex]
Therefore, the correct solution set for the inequality [tex]\(3^{2x - 1} < 27\)[/tex] is [tex]\((-∞, 2)\)[/tex], which corresponds to the first choice in the list given:
[tex]\[ (-\infty, 2) \][/tex]
