Algebra Examples

${(\frac{1}{81})}^{3 - x} \leq 9$

Step 1

Apply the product rule to $\frac{1}{81}$ .

$\frac{1^{3 - x}}{81^{3 - x}} \leq 9$

Step 2

One to any power is one.

$\frac{1}{81^{3 - x}} \leq 9$

Step 3

Move $81^{3 - x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$81^{- (3 - x)} \leq 9$

Step 4

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

$9^{2 (- (3 - x))} \leq 9^{1}$

Step 5

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

$2 (- (3 - x)) \leq 1$

Step 6

Solve for $x$ .

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Step 6.1

Simplify.

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Step 6.1.1

Apply the distributive property.

$2 (- 1 \cdot 3 - - x) \leq 1$

Step 6.1.2

Multiply $- 1$ by $3$ .

$2 (- 3 - - x) \leq 1$

Step 6.1.3

Multiply $- - x$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$2 (- 3 + 1 x) \leq 1$

Step 6.1.3.2

Multiply $x$ by $1$ .

$2 (- 3 + x) \leq 1$

Step 6.2

Divide each term in $2 (- 3 + x) \leq 1$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 (- 3 + x) \leq 1$ by $2$ .

$\frac{2 (- 3 + x)}{2} \leq \frac{1}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 (- 3 + x)}{2} \leq \frac{1}{2}$

Step 6.2.2.1.2

Divide $- 3 + x$ by $1$ .

$- 3 + x \leq \frac{1}{2}$

Step 6.3

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

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Step 6.3.1

Add $3$ to both sides of the inequality.

$x \leq \frac{1}{2} + 3$

Step 6.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \leq \frac{1}{2} + 3 \cdot \frac{2}{2}$

Step 6.3.3

Combine $3$ and $\frac{2}{2}$ .

$x \leq \frac{1}{2} + \frac{3 \cdot 2}{2}$

Step 6.3.4

Combine the numerators over the common denominator.

$x \leq \frac{1 + 3 \cdot 2}{2}$

Step 6.3.5

Simplify the numerator.

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Step 6.3.5.1

Multiply $3$ by $2$ .

$x \leq \frac{1 + 6}{2}$

Step 6.3.5.2

Add $1$ and $6$ .

$x \leq \frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{7}{2}$

Interval Notation:

$(- \infty, \frac{7}{2}]$

Step 8