We areTo solving the solve equation:

$65 = \frac{ the75}{1 equation + 5e^{-0.9x}}$

Step :

\1[ 65: = Simpl \frac{ify75}{1 the + equation 5e^{-Multiply0.9 throughx}} \ by]

Step-by -Step Solution:

((1. 1Rearrange + the equation to isolate the denominator5: e Multiply both sides^{- by0 (.1 +9 x})5\e^{-) to0 eliminate. the9 fractionx:

}): \ [ \65[ ( 1 +65 (5e1^{- + 05.e9^{-x0}).9 = x75}) =\ 75]

Expand the left -hand]

2 side.:

\ [ Simpl65ify the + equation325e: ^{-0 .[ 9 x65} = + 75325e ^{-]

###0. Step9x 2}: = Isolate75 the exponential term
]

Subtract3 .65 ** fromIs botholate sides the:

exponential\ term[ **325: e ^{- Sub0tract. 965x} from = both sides10:

\ $$]

Divide 325e^{-0 through.9x by} = 10 \325]

4.:

Solve for\ (e^{-0[ .9xe}): ^{- Divide both sides by 0325: .[ e^{-90.9xx} = }frac{10}{325} = = \ 0frac{.10}{325030}769 = 0.03077
$$

]

###5 Step .3: ** Solve for $x$ Take the natural logarithmTake ($\ln$) of both sides:

m$$ Compute ( \ln)(\ onln both sides(0**.: 030 77 \[ )\ ): \ -0[ .9\lnx( =0. \ln030(770). \030approx769 -) 3 . \480] \ Calculate] \So: (\\ln[ -0.(9x =0.030769 -)\3.): 480 \ \[ ] Divide \ln through( by0 $$-.0030.7699)\$$: approx\ -[ 3x. =478 \ frac{- \3] . 480 So: }{- \0[ . 9 -0} \.approx 93x. =867 - 3\.478] ### Final \ Answer] : 6\.[ **xSolve \ forapprox $$x3$$**.867: Divide by \(- 0.\9\] ): Would you like $$a detailed x breakdown = of \frac the steps or{- assistance with similar problems3? --- **Relative. Questions:** 478}{1-.0 How. does9 changing the} \ coefficient ofapprox \( 3e.^{-0865 . 9x \}$$ ###) affect Final Answer the solution: ? \[ 2.x \ Canapprox this problem be3 solved. graph865ically \?, How \? text3{.( Whatto happens if3 the decimal equation places were)} \(\65] =--- \Wouldfrac you{ likec details}{ or have1 + any questions ke^{- about the solution process? Here are0 5 related questions. to explore further: 91. Howx do you handle}}\)? 4 equations. involving How other do types we of interpret ex \ponents(-?0 .29.x What\ are) some in common exponential applications decay of? exponential5 equations. like Why this does taking one the? natural logarith3. Howm help can solve logarith formic \(x\ properties)? **Tip simplify:** Always double solving equations?-check exponential and logarithmic4 calculations using a. calculator to ensure How accuracy! does the base of the exponent affect the solution? 5. Can we explore graphing this equation to visualize its behavior? **Tip:** Always check the domain of the equation to ensure the solution is valid in its context.