Algebra Examples

${(\frac{1}{8})}^{x - 6} < 4^{4 x + 5}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$\frac{1^{x - 6}}{8^{x - 6}} < 4^{4 x + 5}$

Step 2

One to any power is one.

$\frac{1}{8^{x - 6}} < 4^{4 x + 5}$

Step 3

Move $8^{x - 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{- (x - 6)} < 4^{4 x + 5}$

Step 4

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

$2^{3 (- (x - 6))} < 2^{2 (4 x + 5)}$

Step 5

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

$3 (- (x - 6)) < 2 (4 x + 5)$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Rewrite.

$0 + 0 + 3 (- (x - 6)) < 2 (4 x + 5)$

Step 6.1.2

Simplify by adding zeros.

$3 (- (x - 6)) < 2 (4 x + 5)$

Step 6.1.3

Apply the distributive property.

$3 (- x - - 6) < 2 (4 x + 5)$

Step 6.1.4

Multiply $- 1$ by $- 6$ .

$3 (- x + 6) < 2 (4 x + 5)$

Step 6.1.5

Apply the distributive property.

$3 (- x) + 3 \cdot 6 < 2 (4 x + 5)$

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 1$ by $3$ .

$- 3 x + 3 \cdot 6 < 2 (4 x + 5)$

Step 6.1.6.2

Multiply $3$ by $6$ .

$- 3 x + 18 < 2 (4 x + 5)$

Step 6.2

Simplify $2 (4 x + 5)$ .

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Step 6.2.1

Apply the distributive property.

$- 3 x + 18 < 2 (4 x) + 2 \cdot 5$

Step 6.2.2

Multiply.

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Step 6.2.2.1

Multiply $4$ by $2$ .

$- 3 x + 18 < 8 x + 2 \cdot 5$

Step 6.2.2.2

Multiply $2$ by $5$ .

$- 3 x + 18 < 8 x + 10$

Step 6.3

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

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Step 6.3.1

Subtract $8 x$ from both sides of the inequality.

$- 3 x + 18 - 8 x < 10$

Step 6.3.2

Subtract $8 x$ from $- 3 x$ .

$- 11 x + 18 < 10$

Step 6.4

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

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Step 6.4.1

Subtract $18$ from both sides of the inequality.

$- 11 x < 10 - 18$

Step 6.4.2

Subtract $18$ from $10$ .

$- 11 x < - 8$

Step 6.5

Divide each term in $- 11 x < - 8$ by $- 11$ and simplify.

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Step 6.5.1

Divide each term in $- 11 x < - 8$ by $- 11$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 11 x}{- 11} > \frac{- 8}{- 11}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 11$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} > \frac{- 8}{- 11}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 8}{- 11}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Dividing two negative values results in a positive value.

$x > \frac{8}{11}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{8}{11}$

Interval Notation:

$(\frac{8}{11}, \infty)$

Step 8