Algebra Examples

$14^{5 x + 2} < 11^{4 x - 1}$

Step 1

Solve $x < - \frac{\ln (11) + 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$ .

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Step 1.1

Take the log of both sides of the inequality.

$\ln (14^{5 x + 2}) < \ln (11^{4 x - 1})$

Step 1.2

Expand $\ln (14^{5 x + 2})$ by moving $5 x + 2$ outside the logarithm.

$(5 x + 2) \ln (14) < \ln (11^{4 x - 1})$

Step 1.3

Expand $\ln (11^{4 x - 1})$ by moving $4 x - 1$ outside the logarithm.

$(5 x + 2) \ln (14) < (4 x - 1) \ln (11)$

Step 1.4

Solve the inequality for $x$ .

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Step 1.4.1

Simplify $(5 x + 2) \ln (14)$ .

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Step 1.4.1.1

Rewrite.

$0 + 0 + (5 x + 2) \ln (14) < (4 x - 1) \ln (11)$

Step 1.4.1.2

Simplify by adding zeros.

$(5 x + 2) \ln (14) < (4 x - 1) \ln (11)$

Step 1.4.1.3

Apply the distributive property.

$5 x \ln (14) + 2 \ln (14) < (4 x - 1) \ln (11)$

Step 1.4.2

Simplify $(4 x - 1) \ln (11)$ .

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Step 1.4.2.1

Apply the distributive property.

$5 x \ln (14) + 2 \ln (14) < 4 x \ln (11) - 1 \ln (11)$

Step 1.4.2.2

Rewrite $- 1 \ln (11)$ as $- \ln (11)$ .

$5 x \ln (14) + 2 \ln (14) < 4 x \ln (11) - \ln (11)$

Step 1.4.3

Subtract $4 x \ln (11)$ from both sides of the inequality.

$5 x \ln (14) + 2 \ln (14) - 4 x \ln (11) < - \ln (11)$

Step 1.4.4

Subtract $2 \ln (14)$ from both sides of the inequality.

$5 x \ln (14) - 4 x \ln (11) < - \ln (11) - 2 \ln (14)$

Step 1.4.5

Factor $x$ out of $5 x \ln (14) - 4 x \ln (11)$ .

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Step 1.4.5.1

Factor $x$ out of $5 x \ln (14)$ .

$x (5 \ln (14)) - 4 x \ln (11) < - \ln (11) - 2 \ln (14)$

Step 1.4.5.2

Factor $x$ out of $- 4 x \ln (11)$ .

$x (5 \ln (14)) + x (- 4 \ln (11)) < - \ln (11) - 2 \ln (14)$

Step 1.4.5.3

Factor $x$ out of $x (5 \ln (14)) + x (- 4 \ln (11))$ .

$x (5 \ln (14) - 4 \ln (11)) < - \ln (11) - 2 \ln (14)$

Step 1.4.6

Divide each term in $x (5 \ln (14) - 4 \ln (11)) < - \ln (11) - 2 \ln (14)$ by $5 \ln (14) - 4 \ln (11)$ and simplify.

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Step 1.4.6.1

Divide each term in $x (5 \ln (14) - 4 \ln (11)) < - \ln (11) - 2 \ln (14)$ by $5 \ln (14) - 4 \ln (11)$ .

$\frac{x (5 \ln (14) - 4 \ln (11))}{5 \ln (14) - 4 \ln (11)} < \frac{- \ln (11)}{5 \ln (14) - 4 \ln (11)} + \frac{- 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.2

Simplify the left side.

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Step 1.4.6.2.1

Cancel the common factor of $5 \ln (14) - 4 \ln (11)$ .

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Step 1.4.6.2.1.1

Cancel the common factor.

$\frac{x (5 \ln (14) - 4 \ln (11))}{5 \ln (14) - 4 \ln (11)} < \frac{- \ln (11)}{5 \ln (14) - 4 \ln (11)} + \frac{- 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.2.1.2

Divide $x$ by $1$ .

$x < \frac{- \ln (11)}{5 \ln (14) - 4 \ln (11)} + \frac{- 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3

Simplify the right side.

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Step 1.4.6.3.1

Combine the numerators over the common denominator.

$x < \frac{- \ln (11) - 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3.2

Factor $- 1$ out of $- \ln (11)$ .

$x < \frac{- \ln (11) - 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3.3

Factor $- 1$ out of $- 2 \ln (14)$ .

$x < \frac{- \ln (11) - (2 \ln (14))}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3.4

Factor $- 1$ out of $- \ln (11) - (2 \ln (14))$ .

$x < \frac{- (\ln (11) + 2 \ln (14))}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3.5

Simplify the expression.

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Step 1.4.6.3.5.1

Rewrite $- (\ln (11) + 2 \ln (14))$ as $- 1 (\ln (11) + 2 \ln (14))$ .

$x < \frac{- 1 (\ln (11) + 2 \ln (14))}{5 \ln (14) - 4 \ln (11)}$

Step 1.4.6.3.5.2

Move the negative in front of the fraction.

$x < - \frac{\ln (11) + 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$

Step 2

Use the inequality $x < - \frac{\ln (11) + 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}$ to build the set notation.

${x | x < - \frac{\ln (11) + 2 \ln (14)}{5 \ln (14) - 4 \ln (11)}}$

Step 3