Algebra Examples

${(\frac{1}{8})}^{- 2 x - 6} > {(\frac{1}{32})}^{- x + 11}$

Step 1

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

${(\frac{1}{2})}^{3 (- 2 x - 6)} > {(\frac{1}{2})}^{5 (- x + 11)}$

Step 2

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

$3 (- 2 x - 6) > 5 (- x + 11)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (- 2 x - 6)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (- 2 x - 6) > 5 (- x + 11)$

Step 3.1.2

Simplify by adding zeros.

$3 (- 2 x - 6) > 5 (- x + 11)$

Step 3.1.3

Apply the distributive property.

$3 (- 2 x) + 3 \cdot - 6 > 5 (- x + 11)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $- 2$ by $3$ .

$- 6 x + 3 \cdot - 6 > 5 (- x + 11)$

Step 3.1.4.2

Multiply $3$ by $- 6$ .

$- 6 x - 18 > 5 (- x + 11)$

Step 3.2

Simplify $5 (- x + 11)$ .

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Step 3.2.1

Apply the distributive property.

$- 6 x - 18 > 5 (- x) + 5 \cdot 11$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $- 1$ by $5$ .

$- 6 x - 18 > - 5 x + 5 \cdot 11$

Step 3.2.2.2

Multiply $5$ by $11$ .

$- 6 x - 18 > - 5 x + 55$

Step 3.3

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

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Step 3.3.1

Add $5 x$ to both sides of the inequality.

$- 6 x - 18 + 5 x > 55$

Step 3.3.2

Add $- 6 x$ and $5 x$ .

$- x - 18 > 55$

Step 3.4

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

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Step 3.4.1

Add $18$ to both sides of the inequality.

$- x > 55 + 18$

Step 3.4.2

Add $55$ and $18$ .

$- x > 73$

Step 3.5

Divide each term in $- x > 73$ by $- 1$ and simplify.

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Step 3.5.1

Divide each term in $- x > 73$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{73}{- 1}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{73}{- 1}$

Step 3.5.2.2

Divide $x$ by $1$ .

$x < \frac{73}{- 1}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $73$ by $- 1$ .

$x < - 73$

Step 4

The solution consists of all of the true intervals.

$x > - 73$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > - 73$

Interval Notation:

$(- 73, \infty)$

Step 6