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Carlie generates the table of values shown below in order to graph and solve the inequality [tex]\( 3^{2x-1} \ \textless \ 27 \)[/tex].

[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & \frac{1}{3} \\
\hline
1 & 3 \\
\hline
2 & 27 \\
\hline
3 & 243 \\
\hline
\end{array}
\][/tex]

What is the solution set for the exponential inequality?

A. [tex]\((- \infty, 2)\)[/tex]

B. [tex]\((2, \infty)\)[/tex]

C. [tex]\((-2, 2)\)[/tex]

D. [tex]\((- \infty, 2) \cup (2, \infty)\)[/tex]



Answer :

GinnyAnswer
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]

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