Precalculus Examples

$g (x) = 2 e^{x} + 6 e^{- x} - 7$

Step 1

Set $2 e^{x} + 6 e^{- x} - 7$ equal to $0$ .

$2 e^{x} + 6 e^{- x} - 7 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite $e^{- x}$ as exponentiation.

$2 e^{x} + 6 {(e^{x})}^{- 1} - 7 = 0$

Step 2.2

Substitute $u$ for $e^{x}$ .

$2 u + 6 u^{- 1} - 7 = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 u + 6 (\frac{1}{u}) - 7 = 0$

Step 2.3.2

Combine $6$ and $\frac{1}{u}$ .

$2 u + \frac{6}{u} - 7 = 0$

Step 2.4

Solve for $u$ .

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Step 2.4.1

Find the LCD of the terms in the equation.

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Step 2.4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1, 1$

Step 2.4.1.2

The LCM of one and any expression is the expression.

$u$

Step 2.4.2

Multiply each term in $2 u + \frac{6}{u} - 7 = 0$ by $u$ to eliminate the fractions.

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Step 2.4.2.1

Multiply each term in $2 u + \frac{6}{u} - 7 = 0$ by $u$ .

$2 u \cdot u + \frac{6}{u} u - 7 u = 0 u$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Simplify each term.

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Step 2.4.2.2.1.1

Multiply $u$ by $u$ by adding the exponents.

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Step 2.4.2.2.1.1.1

Move $u$ .

$2 (u \cdot u) + \frac{6}{u} u - 7 u = 0 u$

Step 2.4.2.2.1.1.2

Multiply $u$ by $u$ .

$2 u^{2} + \frac{6}{u} u - 7 u = 0 u$

Step 2.4.2.2.1.2

Cancel the common factor of $u$ .

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Step 2.4.2.2.1.2.1

Cancel the common factor.

$2 u^{2} + \frac{6}{u} u - 7 u = 0 u$

Step 2.4.2.2.1.2.2

Rewrite the expression.

$2 u^{2} + 6 - 7 u = 0 u$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Multiply $0$ by $u$ .

$2 u^{2} + 6 - 7 u = 0$

Step 2.4.3

Solve the equation.

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Step 2.4.3.1

Factor by grouping.

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Step 2.4.3.1.1

Reorder terms.

$2 u^{2} - 7 u + 6 = 0$

Step 2.4.3.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 6 = 12$ and whose sum is $b = - 7$ .

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Step 2.4.3.1.2.1

Factor $- 7$ out of $- 7 u$ .

$2 u^{2} - 7 u + 6 = 0$

Step 2.4.3.1.2.2

Rewrite $- 7$ as $- 3$ plus $- 4$

$2 u^{2} + (- 3 - 4) u + 6 = 0$

Step 2.4.3.1.2.3

Apply the distributive property.

$2 u^{2} - 3 u - 4 u + 6 = 0$

Step 2.4.3.1.3

Factor out the greatest common factor from each group.

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Step 2.4.3.1.3.1

Group the first two terms and the last two terms.

$(2 u^{2} - 3 u) - 4 u + 6 = 0$

Step 2.4.3.1.3.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u - 3) - 2 (2 u - 3) = 0$

Step 2.4.3.1.4

Factor the polynomial by factoring out the greatest common factor, $2 u - 3$ .

$(2 u - 3) (u - 2) = 0$

Step 2.4.3.2

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

$2 u - 3 = 0$

$u - 2 = 0$

Step 2.4.3.3

Set $2 u - 3$ equal to $0$ and solve for $u$ .

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Step 2.4.3.3.1

Set $2 u - 3$ equal to $0$ .

$2 u - 3 = 0$

Step 2.4.3.3.2

Solve $2 u - 3 = 0$ for $u$ .

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Step 2.4.3.3.2.1

Add $3$ to both sides of the equation.

$2 u = 3$

Step 2.4.3.3.2.2

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

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Step 2.4.3.3.2.2.1

Divide each term in $2 u = 3$ by $2$ .

$\frac{2 u}{2} = \frac{3}{2}$

Step 2.4.3.3.2.2.2

Simplify the left side.

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Step 2.4.3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{3}{2}$

Step 2.4.3.3.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{3}{2}$

Step 2.4.3.4

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 2.4.3.4.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 2.4.3.4.2

Add $2$ to both sides of the equation.

$u = 2$

Step 2.4.3.5

The final solution is all the values that make $(2 u - 3) (u - 2) = 0$ true.

$u = \frac{3}{2}, 2$

Step 2.5

Substitute $\frac{3}{2}$ for $u$ in $u = e^{x}$ .

$\frac{3}{2} = e^{x}$

Step 2.6

Solve $\frac{3}{2} = e^{x}$ .

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Step 2.6.1

Rewrite the equation as $e^{x} = \frac{3}{2}$ .

$e^{x} = \frac{3}{2}$

Step 2.6.2

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

$\ln (e^{x}) = \ln (\frac{3}{2})$

Step 2.6.3

Expand the left side.

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Step 2.6.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (\frac{3}{2})$

Step 2.6.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (\frac{3}{2})$

Step 2.6.3.3

Multiply $x$ by $1$ .

$x = \ln (\frac{3}{2})$

Step 2.7

Substitute $2$ for $u$ in $u = e^{x}$ .

$2 = e^{x}$

Step 2.8

Solve $2 = e^{x}$ .

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Step 2.8.1

Rewrite the equation as $e^{x} = 2$ .

$e^{x} = 2$

Step 2.8.2

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

$\ln (e^{x}) = \ln (2)$

Step 2.8.3

Expand the left side.

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Step 2.8.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (2)$

Step 2.8.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (2)$

Step 2.8.3.3

Multiply $x$ by $1$ .

$x = \ln (2)$

Step 2.9

List the solutions that makes the equation true.

$x = \ln (\frac{3}{2}), \ln (2)$

Step 3