Algebra Examples

$4 {(\frac{1}{3})}^{x} < 36$

Step 1

Divide each term in $4 {(\frac{1}{3})}^{x} < 36$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 {(\frac{1}{3})}^{x} < 36$ by $4$ .

$\frac{4 {(\frac{1}{3})}^{x}}{4} < \frac{36}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 {(\frac{1}{3})}^{x}}{4} < \frac{36}{4}$

Step 1.2.1.2

Divide ${(\frac{1}{3})}^{x}$ by $1$ .

${(\frac{1}{3})}^{x} < \frac{36}{4}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{x}}{3^{x}} < \frac{36}{4}$

Step 1.2.2.2

One to any power is one.

$\frac{1}{3^{x}} < \frac{36}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $36$ by $4$ .

$\frac{1}{3^{x}} < 9$

Step 2

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

$3^{- x} < 9$

Step 3

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

$3^{- x} < 3^{2}$

Step 4

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

$- x < 2$

Step 5

Divide each term in $- x < 2$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $2$ by $- 1$ .

$x > - 2$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > - 2$

Interval Notation:

$(- 2, \infty)$

Step 7