Precalculus Examples

$9^{x + 1} = 27^{3 - x}$

Step 1

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

$3^{2 (x + 1)} = 3^{3 (3 - x)}$

Step 2

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

$2 (x + 1) = 3 (3 - x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (x + 1)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (x + 1) = 3 (3 - x)$

Step 3.1.2

Simplify by adding zeros.

$2 (x + 1) = 3 (3 - x)$

Step 3.1.3

Apply the distributive property.

$2 x + 2 \cdot 1 = 3 (3 - x)$

Step 3.1.4

Multiply $2$ by $1$ .

$2 x + 2 = 3 (3 - x)$

Step 3.2

Simplify $3 (3 - x)$ .

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Step 3.2.1

Apply the distributive property.

$2 x + 2 = 3 \cdot 3 + 3 (- x)$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $3$ by $3$ .

$2 x + 2 = 9 + 3 (- x)$

Step 3.2.2.2

Multiply $- 1$ by $3$ .

$2 x + 2 = 9 - 3 x$

Step 3.3

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

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Step 3.3.1

Add $3 x$ to both sides of the equation.

$2 x + 2 + 3 x = 9$

Step 3.3.2

Add $2 x$ and $3 x$ .

$5 x + 2 = 9$

Step 3.4

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

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Step 3.4.1

Subtract $2$ from both sides of the equation.

$5 x = 9 - 2$

Step 3.4.2

Subtract $2$ from $9$ .

$5 x = 7$

Step 3.5

Divide each term in $5 x = 7$ by $5$ and simplify.

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Step 3.5.1

Divide each term in $5 x = 7$ by $5$ .

$\frac{5 x}{5} = \frac{7}{5}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $5$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{7}{5}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{5}$

Decimal Form:

$x = 1.4$

Mixed Number Form:

$x = 1 \frac{2}{5}$