Solve for x e^(x-2)=e^(3x+1)

$e^{x - 2} = e^{3 x + 1}$

Create equivalent expressions in the equation that all have equal bases.

$e^{x - 2} = e^{3 x + 1}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x - 2 = 3 x + 1$

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$x - 2 - 3 x = 1$

Subtract $3 x$ from $x$ .

$- 2 x - 2 = 1$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$- 2 x = 1 + 2$

Add $1$ and $2$ .

$- 2 x = 3$

Divide each term in $- 2 x = 3$ by $- 2$ and simplify.

Divide each term in $- 2 x = 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Divide $x$ by $1$ .

$x = \frac{3}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$