Precalculus Examples

$- x e^{- x} + e^{- x} = 0$

Step 1

Factor $e^{- x}$ out of $- x e^{- x} + e^{- x}$ .

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Step 1.1

Factor $e^{- x}$ out of $- x e^{- x}$ .

$e^{- x} (- x) + e^{- x} = 0$

Step 1.2

Multiply by $1$ .

$e^{- x} (- x) + e^{- x} \cdot 1 = 0$

Step 1.3

Factor $e^{- x}$ out of $e^{- x} (- x) + e^{- x} \cdot 1$ .

$e^{- x} (- x + 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{- x} = 0$

$- x + 1 = 0$

Step 3

Set $e^{- x}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $e^{- x}$ equal to $0$ .

$e^{- x} = 0$

Step 3.2

Solve $e^{- x} = 0$ for $x$ .

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Step 3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x}) = \ln (0)$

Step 3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 3.2.3

There is no solution for $e^{- x} = 0$

No solution

Step 4

Set $- x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $- x + 1$ equal to $0$ .

$- x + 1 = 0$

Step 4.2

Solve $- x + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 4.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2.2.2

Divide $x$ by $1$ .

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

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 5

The final solution is all the values that make $e^{- x} (- x + 1) = 0$ true.

$x = 1$