Questions: Use the ALEKS graphing calculator to solve the equation. e^(2x-2)=3x+1 Round to the nearest hundredth. If there is more than one solution, separate them with commas. x=

Solution Steps

To solve the equation \( e^{2x-2} = 3x + 1 \), we need to find the value(s) of \( x \) where both sides of the equation are equal. This is a transcendental equation, which typically requires numerical methods to solve. We can use Python's numerical solvers, such as those in the scipy.optimize library, to find the roots of the equation. We'll define a function representing the difference between the two sides of the equation and use a root-finding method to solve for \( x \).

We start with the equation given in the problem:

\[ e^{2x - 2} = 3x + 1 \]

To find the roots, we can rearrange the equation to form a function:

\[ f(x) = e^{2x - 2} - (3x + 1) \]

We need to find the value(s) of \( x \) for which \( f(x) = 0 \).

Using numerical methods, we find that the solution to the equation is approximately:

\[ x \approx -0.309018 \]

Rounding this value to the nearest hundredth gives us:

\[ x \approx -0.31 \]

Final Answer