Algebra Examples

${(\frac{3}{4})}^{x^{2} + 5 x} > {(\frac{4}{3})}^{3 x}$

Step 1

Take the log of both sides of the inequality.

$\ln ({(\frac{3}{4})}^{x^{2} + 5 x}) > \ln ({(\frac{4}{3})}^{3 x})$

Step 2

Expand $\ln ({(\frac{3}{4})}^{x^{2} + 5 x})$ by moving $x^{2} + 5 x$ outside the logarithm.

$(x^{2} + 5 x) \ln (\frac{3}{4}) > \ln ({(\frac{4}{3})}^{3 x})$

Step 3

Rewrite $\ln (\frac{3}{4})$ as $\ln (3) - \ln (4)$ .

$(x^{2} + 5 x) (\ln (3) - \ln (4)) > \ln ({(\frac{4}{3})}^{3 x})$

Step 4

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$(x^{2} + 5 x) (\ln (3) - \ln (2^{2})) > \ln ({(\frac{4}{3})}^{3 x})$

Step 5

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

$(x^{2} + 5 x) (\ln (3) - (2 \ln (2))) > \ln ({(\frac{4}{3})}^{3 x})$

Step 6

Multiply $2$ by $- 1$ .

$(x^{2} + 5 x) (\ln (3) - 2 \ln (2)) > \ln ({(\frac{4}{3})}^{3 x})$

Step 7

Expand $\ln ({(\frac{4}{3})}^{3 x})$ by moving $3 x$ outside the logarithm.

$(x^{2} + 5 x) (\ln (3) - 2 \ln (2)) > 3 x \ln (\frac{4}{3})$

Step 8

Rewrite $\ln (\frac{4}{3})$ as $\ln (4) - \ln (3)$ .

$(x^{2} + 5 x) (\ln (3) - 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9

Solve the inequality for $x$ .

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Step 9.1

Simplify $(x^{2} + 5 x) (\ln (3) - 2 \ln (2))$ .

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Step 9.1.1

Rewrite.

$0 + 0 + (x^{2} + 5 x) (\ln (3) - 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.2

Simplify by adding zeros.

$(x^{2} + 5 x) (\ln (3) - 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.3

Expand $(x^{2} + 5 x) (\ln (3) - 2 \ln (2))$ using the FOIL Method.

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Step 9.1.3.1

Apply the distributive property.

$x^{2} (\ln (3) - 2 \ln (2)) + 5 x (\ln (3) - 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.3.2

Apply the distributive property.

$x^{2} \ln (3) + x^{2} (- 2 \ln (2)) + 5 x (\ln (3) - 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.3.3

Apply the distributive property.

$x^{2} \ln (3) + x^{2} (- 2 \ln (2)) + 5 x \ln (3) + 5 x (- 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.4

Simplify each term.

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Step 9.1.4.1

Rewrite using the commutative property of multiplication.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) + 5 x (- 2 \ln (2)) > 3 x (\ln (4) - \ln (3))$

Step 9.1.4.2

Rewrite using the commutative property of multiplication.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) + 5 \cdot - 2 x \ln (2) > 3 x (\ln (4) - \ln (3))$

Step 9.1.4.3

Multiply $5$ by $- 2$ .

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) > 3 x (\ln (4) - \ln (3))$

Step 9.2

Simplify $3 x (\ln (4) - \ln (3))$ .

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Step 9.2.1

Apply the distributive property.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) > 3 x \ln (4) + 3 x (- \ln (3))$

Step 9.2.2

Simplify the expression.

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Step 9.2.2.1

Rewrite using the commutative property of multiplication.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) > 3 x \ln (4) + 3 \cdot - 1 x \ln (3)$

Step 9.2.2.2

Multiply $3$ by $- 1$ .

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) > 3 x \ln (4) - 3 x \ln (3)$

Step 9.3

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

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Step 9.3.1

Subtract $3 x \ln (4)$ from both sides of the inequality.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) - 3 x \ln (4) > - 3 x \ln (3)$

Step 9.3.2

Add $3 x \ln (3)$ to both sides of the inequality.

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 5 x \ln (3) - 10 x \ln (2) - 3 x \ln (4) + 3 x \ln (3) > 0$

Step 9.3.3

Add $5 x \ln (3)$ and $3 x \ln (3)$ .

$x^{2} \ln (3) - 2 x^{2} \ln (2) + 8 x \ln (3) - 10 x \ln (2) - 3 x \ln (4) > 0$

Step 9.4

Factor $x$ out of $x^{2} \ln (3) - 2 x^{2} \ln (2) + 8 x \ln (3) - 10 x \ln (2) - 3 x \ln (4)$ .

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Step 9.4.1

Factor $x$ out of $x^{2} \ln (3)$ .

$x (x \ln (3)) - 2 x^{2} \ln (2) + 8 x \ln (3) - 10 x \ln (2) - 3 x \ln (4) > 0$

Step 9.4.2

Factor $x$ out of $- 2 x^{2} \ln (2)$ .

$x (x \ln (3)) + x (- 2 x \ln (2)) + 8 x \ln (3) - 10 x \ln (2) - 3 x \ln (4) > 0$

Step 9.4.3

Factor $x$ out of $8 x \ln (3)$ .

$x (x \ln (3)) + x (- 2 x \ln (2)) + x (8 \ln (3)) - 10 x \ln (2) - 3 x \ln (4) > 0$

Step 9.4.4

Factor $x$ out of $- 10 x \ln (2)$ .

$x (x \ln (3)) + x (- 2 x \ln (2)) + x (8 \ln (3)) + x (- 10 \ln (2)) - 3 x \ln (4) > 0$

Step 9.4.5

Factor $x$ out of $- 3 x \ln (4)$ .

$x (x \ln (3)) + x (- 2 x \ln (2)) + x (8 \ln (3)) + x (- 10 \ln (2)) + x (- 3 \ln (4)) > 0$

Step 9.4.6

Factor $x$ out of $x (x \ln (3)) + x (- 2 x \ln (2))$ .

$x (x \ln (3) - 2 x \ln (2)) + x (8 \ln (3)) + x (- 10 \ln (2)) + x (- 3 \ln (4)) > 0$

Step 9.4.7

Factor $x$ out of $x (x \ln (3) - 2 x \ln (2)) + x (8 \ln (3))$ .

$x (x \ln (3) - 2 x \ln (2) + 8 \ln (3)) + x (- 10 \ln (2)) + x (- 3 \ln (4)) > 0$

Step 9.4.8

Factor $x$ out of $x (x \ln (3) - 2 x \ln (2) + 8 \ln (3)) + x (- 10 \ln (2))$ .

$x (x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2)) + x (- 3 \ln (4)) > 0$

Step 9.4.9

Factor $x$ out of $x (x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2)) + x (- 3 \ln (4))$ .

$x (x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4)) > 0$

Step 9.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4) = 0$

Step 9.6

Set $x$ equal to $0$ .

$x = 0$

Step 9.7

Set $x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4)$ equal to $0$ and solve for $x$ .

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Step 9.7.1

Set $x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4)$ equal to $0$ .

$x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4) = 0$

Step 9.7.2

Solve $x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4) = 0$ for $x$ .

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Step 9.7.2.1

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

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Step 9.7.2.1.1

Subtract $8 \ln (3)$ from both sides of the equation.

$x \ln (3) - 2 x \ln (2) - 10 \ln (2) - 3 \ln (4) = - 8 \ln (3)$

Step 9.7.2.1.2

Add $10 \ln (2)$ to both sides of the equation.

$x \ln (3) - 2 x \ln (2) - 3 \ln (4) = - 8 \ln (3) + 10 \ln (2)$

Step 9.7.2.1.3

Add $3 \ln (4)$ to both sides of the equation.

$x \ln (3) - 2 x \ln (2) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$

Step 9.7.2.2

Factor $x$ out of $x \ln (3) - 2 x \ln (2)$ .

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Step 9.7.2.2.1

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) - 2 x \ln (2) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$

Step 9.7.2.2.2

Factor $x$ out of $- 2 x \ln (2)$ .

$x (\ln (3)) + x (- 2 \ln (2)) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$

Step 9.7.2.2.3

Factor $x$ out of $x (\ln (3)) + x (- 2 \ln (2))$ .

$x (\ln (3) - 2 \ln (2)) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$

Step 9.7.2.3

Divide each term in $x (\ln (3) - 2 \ln (2)) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$ by $\ln (3) - 2 \ln (2)$ and simplify.

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Step 9.7.2.3.1

Divide each term in $x (\ln (3) - 2 \ln (2)) = - 8 \ln (3) + 10 \ln (2) + 3 \ln (4)$ by $\ln (3) - 2 \ln (2)$ .

$\frac{x (\ln (3) - 2 \ln (2))}{\ln (3) - 2 \ln (2)} = \frac{- 8 \ln (3)}{\ln (3) - 2 \ln (2)} + \frac{10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.2

Simplify the left side.

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Step 9.7.2.3.2.1

Cancel the common factor of $\ln (3) - 2 \ln (2)$ .

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Step 9.7.2.3.2.1.1

Cancel the common factor.

$\frac{x (\ln (3) - 2 \ln (2))}{\ln (3) - 2 \ln (2)} = \frac{- 8 \ln (3)}{\ln (3) - 2 \ln (2)} + \frac{10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8 \ln (3)}{\ln (3) - 2 \ln (2)} + \frac{10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3

Simplify the right side.

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Step 9.7.2.3.3.1

Simplify terms.

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Step 9.7.2.3.3.1.1

Move the negative in front of the fraction.

$x = - \frac{8 \ln (3)}{\ln (3) - 2 \ln (2)} + \frac{10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.1.2

Simplify terms.

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Step 9.7.2.3.3.1.2.1

Combine the numerators over the common denominator.

$x = \frac{- 8 \ln (3) + 10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.1.2.2

Factor $2$ out of $- 8 \ln (3) + 10 \ln (2)$ .

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Step 9.7.2.3.3.1.2.2.1

Factor $2$ out of $- 8 \ln (3)$ .

$x = \frac{2 (- 4 \ln (3)) + 10 \ln (2)}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.1.2.2.2

Factor $2$ out of $10 \ln (2)$ .

$x = \frac{2 (- 4 \ln (3)) + 2 (5 \ln (2))}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.1.2.2.3

Factor $2$ out of $2 (- 4 \ln (3)) + 2 (5 \ln (2))$ .

$x = \frac{2 (- 4 \ln (3) + 5 \ln (2))}{\ln (3) - 2 \ln (2)} + \frac{3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.1.2.3

Combine the numerators over the common denominator.

$x = \frac{2 (- 4 \ln (3) + 5 \ln (2)) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.2

Simplify the numerator.

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Step 9.7.2.3.3.2.1

Apply the distributive property.

$x = \frac{2 (- 4 \ln (3)) + 2 (5 \ln (2)) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 8 \ln (3) + 2 (5 \ln (2)) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.2.3

Multiply $5$ by $2$ .

$x = \frac{- 8 \ln (3) + 10 \ln (2) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3

Simplify with factoring out.

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Step 9.7.2.3.3.3.1

Factor $- 1$ out of $- 8 \ln (3)$ .

$x = \frac{- (8 \ln (3)) + 10 \ln (2) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.2

Factor $- 1$ out of $10 \ln (2)$ .

$x = \frac{- (8 \ln (3)) - (- 10 \ln (2)) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.3

Factor $- 1$ out of $- (8 \ln (3)) - (- 10 \ln (2))$ .

$x = \frac{- (8 \ln (3) - 10 \ln (2)) + 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.4

Factor $- 1$ out of $3 \ln (4)$ .

$x = \frac{- (8 \ln (3) - 10 \ln (2)) - (- 3 \ln (4))}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.5

Factor $- 1$ out of $- (8 \ln (3) - 10 \ln (2)) - (- 3 \ln (4))$ .

$x = \frac{- (8 \ln (3) - 10 \ln (2) - 3 \ln (4))}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.6

Simplify the expression.

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Step 9.7.2.3.3.3.6.1

Rewrite $- (8 \ln (3) - 10 \ln (2) - 3 \ln (4))$ as $- 1 (8 \ln (3) - 10 \ln (2) - 3 \ln (4))$ .

$x = \frac{- 1 (8 \ln (3) - 10 \ln (2) - 3 \ln (4))}{\ln (3) - 2 \ln (2)}$

Step 9.7.2.3.3.3.6.2

Move the negative in front of the fraction.

$x = - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 9.8

The final solution is all the values that make $x (x \ln (3) - 2 x \ln (2) + 8 \ln (3) - 10 \ln (2) - 3 \ln (4)) > 0$ true.

$x = 0, - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

Step 10

Use each root to create test intervals.

$x < - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$

$- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$

$x > 0$

Step 11

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 11.1

Test a value on the interval $x < - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$ and see if this value makes the original inequality true.

$x = - 10$

Step 11.1.2

Replace $x$ with $- 10$ in the original inequality.

${(\frac{3}{4})}^{{(- 10)}^{2} + 5 (- 10)} > {(\frac{4}{3})}^{3 (- 10)}$

Step 11.1.3

The left side $5.66321656 E - 7$ is not greater than the right side $0.00017858$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$ and see if this value makes the original inequality true.

$x = - 4$

Step 11.2.2

Replace $x$ with $- 4$ in the original inequality.

${(\frac{3}{4})}^{{(- 4)}^{2} + 5 (- 4)} > {(\frac{4}{3})}^{3 (- 4)}$

Step 11.2.3

The left side $3.16049382$ is greater than the right side $0.03167635$ , which means that the given statement is always true.

True

Step 11.3

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

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Step 11.3.1

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

$x = 2$

Step 11.3.2

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

${(\frac{3}{4})}^{{(2)}^{2} + 5 (2)} > {(\frac{4}{3})}^{3 (2)}$

Step 11.3.3

The left side $0.01781794$ is not greater than the right side $5.61865569$ , which means that the given statement is false.

False

Step 11.4

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

$x < - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$ False

$- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$ True

$x > 0$ False

$x < - \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)}$ False

$- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$ True

$x > 0$ False

Step 12

The solution consists of all of the true intervals.

$- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- \frac{8 \ln (3) - 10 \ln (2) - 3 \ln (4)}{\ln (3) - 2 \ln (2)} < x < 0$

Interval Notation:

$(- \frac{\ln (\frac{6561}{65536})}{\ln (\frac{3}{4})}, 0)$

Step 14