To solve the equation [tex]\( e^{-x} = e^{2x + 6} \)[/tex], let’s go through the steps methodically:
1. Given Equation:
[tex]\[
e^{-x} = e^{2x + 6}
\][/tex]
2. Since the bases of the exponents are the same (both [tex]\( e \)[/tex]), we can equate the exponents:
[tex]\[
-x = 2x + 6
\][/tex]
3. Move all the terms involving [tex]\( x \)[/tex] to one side:
[tex]\[
-x - 2x = 6
\][/tex]
4. Combine like terms:
[tex]\[
-3x = 6
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{6}{-3}
\][/tex]
[tex]\[
x = -2
\][/tex]
6. Check the solution with the given options:
The given options are [tex]\( x = 2 \)[/tex], [tex]\( x = 4 \)[/tex], [tex]\( x = -4 \)[/tex], [tex]\( x = -2 \)[/tex].
The correct solution derived from the equation is [tex]\( x = -2 \)[/tex].
Therefore, the correct answer is:
[tex]\[
x = -2
\][/tex]
