Solve: 0.05x^2.32> 4* 1.05^x-1?

if you meant

0.05*x^(2.32) > 4* 1.05^(x-1)

=== I believe the answer is

x> 7.59508046

=== why

multiple both sides by 20

x^2.32 > 80* 1.05^(x-1)

x^2.32 - 80* 1.05^(x-1)> 0

so find the equal "= " case first,

Then, you can try point above and below this point to determine the correct zone.

y = x^2.32 - 80* 1.05^(x-1)

so you are looking where y > 0 to meet the inequality.

Using Newton Raphson Method

graph the equation to show an area to guess

(or just try points looking for 0 crossings )

for example find f(7.4) and f(7.6)

f(7.4) = -5.419984155

f(7.6) = 0.139427036 , so 7.6 is greater than zero so it will be x> 7.XXXX

so you know one answer for equal point is between 7.4 < x< 7.6

Excel indicates for all negative values of x, the y are undefined.

for real numbers

and the only real answer about 7.5

x> 7.5XXXX

now just use newton raphson method to refine the answer

next guess = prev guess - f(x)/f'(x)

ng = pg - (x^2.32 - 80* 1.05^(x-1))/(2.32*(x^1.32) - 80*ln(1.05)*(1.05)^(x-1) )

pg adjustment New Guess

7.5 -0.095988095 7.595988095

7.595988095 0.000907554 7.595080541

7.595080541 8.13622E-08 7.59508046

7.59508046 4.01327E-15 7.59508046

7.59508046 -6.01991E-15 7.59508046

so according to Excel

x> 7.59508046

Desmos graphing utilities also shows negative value of x, but

everything I read tells me the graph is wrong.

Every time I tried to use Newton Raphson method with this

It says (-5.738)^(2.32) is undefined in real numbers

.I know you cannnot have negative even roots. (-5)^(-1/2) is undefined.

(-5)^(2.32) = (-5)^2*(-5)^(0.32)

Add brackets to show what the exponent of 1.05 is.

Ex.

(1.05^x) - 1 or 1.05^(x-1)