To solve the equation [tex]\( 5^{-2x - 9} = 5^{4x + 15} \)[/tex], follow these steps:
1. Recognize the property of exponents: Since the bases on both sides of the equation are the same, we can equate the exponents. This is due to the property of exponentiation that if [tex]\( a^m = a^n \)[/tex] for [tex]\( a \neq 0 \)[/tex], then [tex]\( m = n \)[/tex].
So, equate the exponents:
[tex]\[
-2x - 9 = 4x + 15
\][/tex]
2. Solve the resulting linear equation: Combine the terms involving [tex]\( x \)[/tex] on one side and the constant terms on the other side.
- First, add [tex]\( 2x \)[/tex] to both sides to move the terms involving [tex]\( x \)[/tex] to the same side:
[tex]\[
-2x - 9 + 2x = 4x + 15 + 2x
\][/tex]
Simplify:
[tex]\[
-9 = 6x + 15
\][/tex]
- Next, subtract 15 from both sides to isolate the constant term on the left:
[tex]\[
-9 - 15 = 6x + 15 - 15
\][/tex]
Simplify:
[tex]\[
-24 = 6x
\][/tex]
- Finally, divide both sides by 6 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{-24}{6} = \frac{6x}{6}
\][/tex]
Simplify:
[tex]\[
x = -4
\][/tex]
Therefore, the solution to the equation [tex]\( 5^{-2x - 9} = 5^{4x + 15} \)[/tex] is
[tex]\[
x = -4
\][/tex]
Check if the value of [tex]\( x \)[/tex] is in the given options:
[tex]\[
x = -3, x = -1, x = -4, x = \frac{1}{5}
\][/tex]
The correct answer is [tex]\( x = -4 \)[/tex].
