Algebra Examples

${(\frac{1}{4})}^{\frac{2 - x}{x}} > 64^{2}$

Step 1

Apply the product rule to $\frac{1}{4}$ .

$\frac{1^{\frac{2 - x}{x}}}{4^{\frac{2 - x}{x}}} > 64^{2}$

Step 2

One to any power is one.

$\frac{1}{4^{\frac{2 - x}{x}}} > 64^{2}$

Step 3

Move $4^{\frac{2 - x}{x}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- \frac{2 - x}{x}} > 64^{2}$

Step 4

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

$4^{- \frac{2 - x}{x}} > 4^{3 \cdot 2}$

Step 5

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

$- \frac{2 - x}{x} > 3 \cdot 2$

Step 6

Solve for $x$ .

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Step 6.1

Simplify the right side.

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Step 6.1.1

Multiply $3$ by $2$ .

$- \frac{2 - x}{x} > 6$

Step 6.2

Subtract $6$ from both sides of the inequality.

$- \frac{2 - x}{x} - 6 > 0$

Step 6.3

Simplify $- \frac{2 - x}{x} - 6$ .

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Step 6.3.1

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{2 - x}{x} - 6 \cdot \frac{x}{x} > 0$

Step 6.3.2

Combine $- 6$ and $\frac{x}{x}$ .

$- \frac{2 - x}{x} + \frac{- 6 x}{x} > 0$

Step 6.3.3

Combine the numerators over the common denominator.

$\frac{- (2 - x) - 6 x}{x} > 0$

Step 6.3.4

Simplify the numerator.

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Step 6.3.4.1

Apply the distributive property.

$\frac{- 1 \cdot 2 - - x - 6 x}{x} > 0$

Step 6.3.4.2

Multiply $- 1$ by $2$ .

$\frac{- 2 - - x - 6 x}{x} > 0$

Step 6.3.4.3

Multiply $- - x$ .

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Step 6.3.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 2 + 1 x - 6 x}{x} > 0$

Step 6.3.4.3.2

Multiply $x$ by $1$ .

$\frac{- 2 + x - 6 x}{x} > 0$

Step 6.3.4.4

Subtract $6 x$ from $x$ .

$\frac{- 2 - 5 x}{x} > 0$

Step 6.3.5

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) - 5 x}{x} > 0$

Step 6.3.6

Factor $- 1$ out of $- 5 x$ .

$\frac{- 1 (2) - (5 x)}{x} > 0$

Step 6.3.7

Factor $- 1$ out of $- 1 (2) - (5 x)$ .

$\frac{- 1 (2 + 5 x)}{x} > 0$

Step 6.3.8

Move the negative in front of the fraction.

$- \frac{2 + 5 x}{x} > 0$

Step 6.4

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

$2 + 5 x = 0$

Step 6.5

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 6.6

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 6.6.1

Divide each term in $5 x = - 2$ by $5$ .

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

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor of $5$ .

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Step 6.6.2.1.1

Cancel the common factor.

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

Step 6.6.2.1.2

Divide $x$ by $1$ .

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

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Move the negative in front of the fraction.

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

Step 6.7

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

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

Step 6.8

Consolidate the solutions.

$x = 0, - \frac{2}{5}$

Step 7

Find the domain of $\frac{1}{4^{\frac{2 - x}{x}}} - 4096$ .

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Step 7.1

Set the denominator in $\frac{2 - x}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 7.2

Set the denominator in $\frac{1}{4^{\frac{2 - x}{x}}}$ equal to $0$ to find where the expression is undefined.

$4^{\frac{2 - x}{x}} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

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

$\ln (4^{\frac{2 - x}{x}}) = \ln (0)$

Step 7.3.2

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

Undefined

Step 7.3.3

There is no solution for $4^{\frac{2 - x}{x}} = 0$

No solution

Step 7.4

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 8

Use each root to create test intervals.

$x < - \frac{2}{5}$

$- \frac{2}{5} < x < 0$

$x > 0$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.1

Test a value on the interval $x < - \frac{2}{5}$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - \frac{2}{5}$ and see if this value makes the original inequality true.

$x = - 3$

Step 9.1.2

Replace $x$ with $- 3$ in the original inequality.

${(\frac{1}{4})}^{\frac{2 - (- 3)}{- 3}} > 64^{2}$

Step 9.1.3

The left side $10.07936839$ is not greater than the right side $4096$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- \frac{2}{5} < x < 0$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- \frac{2}{5} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.2$

Step 9.2.2

Replace $x$ with $- 0.2$ in the original inequality.

${(\frac{1}{4})}^{\frac{2 - (- 0.2)}{- 0.2}} > 64^{2}$

Step 9.2.3

The left side $4194304$ is greater than the right side $4096$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $x > 0$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 9.3.2

Replace $x$ with $2$ in the original inequality.

${(\frac{1}{4})}^{\frac{2 - (2)}{2}} > 64^{2}$

Step 9.3.3

The left side $1$ is not greater than the right side $4096$ , which means that the given statement is false.

False

Step 9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ True

$x > 0$ False

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ True

$x > 0$ False

Step 10

The solution consists of all of the true intervals.

$- \frac{2}{5} < x < 0$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{2}{5} < x < 0$

Interval Notation:

$(- \frac{2}{5}, 0)$

Step 12