Algebra Examples

$100^{\frac{3 - 4 x}{2 x + 1}} \leq {0.01}^{\frac{5 x - 8}{2 x + 1}}$

Step 1

Take the log of both sides of the inequality.

$\ln (100^{\frac{3 - 4 x}{2 x + 1}}) \leq \ln ({0.01}^{\frac{5 x - 8}{2 x + 1}})$

Step 2

Expand $\ln (100^{\frac{3 - 4 x}{2 x + 1}})$ by moving $\frac{3 - 4 x}{2 x + 1}$ outside the logarithm.

$\frac{3 - 4 x}{2 x + 1} \ln (100) \leq \ln ({0.01}^{\frac{5 x - 8}{2 x + 1}})$

Step 3

Combine $\frac{3 - 4 x}{2 x + 1}$ and $\ln (100)$ .

$\frac{(3 - 4 x) \ln (100)}{2 x + 1} \leq \ln ({0.01}^{\frac{5 x - 8}{2 x + 1}})$

Step 4

Expand $\ln ({0.01}^{\frac{5 x - 8}{2 x + 1}})$ by moving $\frac{5 x - 8}{2 x + 1}$ outside the logarithm.

$\frac{(3 - 4 x) \ln (100)}{2 x + 1} \leq \frac{5 x - 8}{2 x + 1} \ln (0.01)$

Step 5

Combine $\frac{5 x - 8}{2 x + 1}$ and $\ln (0.01)$ .

$\frac{(3 - 4 x) \ln (100)}{2 x + 1} \leq \frac{(5 x - 8) \ln (0.01)}{2 x + 1}$

Step 6

Solve the inequality for $x$ .

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Step 6.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$(3 - 4 x) \ln (100) \leq (5 x - 8) \ln (0.01)$

Step 6.2

Simplify $(3 - 4 x) \ln (100)$ .

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Step 6.2.1

Rewrite.

$0 + 0 + (3 - 4 x) \ln (100) \leq (5 x - 8) \ln (0.01)$

Step 6.2.2

Simplify by adding zeros.

$(3 - 4 x) \ln (100) \leq (5 x - 8) \ln (0.01)$

Step 6.2.3

Apply the distributive property.

$3 \ln (100) - 4 x \ln (100) \leq (5 x - 8) \ln (0.01)$

Step 6.3

Apply the distributive property.

$3 \ln (100) - 4 x \ln (100) \leq 5 x \ln (0.01) - 8 \ln (0.01)$

Step 6.4

Subtract $5 x \ln (0.01)$ from both sides of the inequality.

$3 \ln (100) - 4 x \ln (100) - 5 x \ln (0.01) \leq - 8 \ln (0.01)$

Step 6.5

Subtract $3 \ln (100)$ from both sides of the inequality.

$- 4 x \ln (100) - 5 x \ln (0.01) \leq - 8 \ln (0.01) - 3 \ln (100)$

Step 6.6

Factor $x$ out of $- 4 x \ln (100) - 5 x \ln (0.01)$ .

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Step 6.6.1

Factor $x$ out of $- 4 x \ln (100)$ .

$x (- 4 \ln (100)) - 5 x \ln (0.01) \leq - 8 \ln (0.01) - 3 \ln (100)$

Step 6.6.2

Factor $x$ out of $- 5 x \ln (0.01)$ .

$x (- 4 \ln (100)) + x (- 5 \ln (0.01)) \leq - 8 \ln (0.01) - 3 \ln (100)$

Step 6.6.3

Factor $x$ out of $x (- 4 \ln (100)) + x (- 5 \ln (0.01))$ .

$x (- 4 \ln (100) - 5 \ln (0.01)) \leq - 8 \ln (0.01) - 3 \ln (100)$

Step 6.7

Divide each term in $x (- 4 \ln (100) - 5 \ln (0.01)) \leq - 8 \ln (0.01) - 3 \ln (100)$ by $- 4 \ln (100) - 5 \ln (0.01)$ and simplify.

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Step 6.7.1

Divide each term in $x (- 4 \ln (100) - 5 \ln (0.01)) \leq - 8 \ln (0.01) - 3 \ln (100)$ by $- 4 \ln (100) - 5 \ln (0.01)$ .

$\frac{x (- 4 \ln (100) - 5 \ln (0.01))}{- 4 \ln (100) - 5 \ln (0.01)} \leq \frac{- 8 \ln (0.01)}{- 4 \ln (100) - 5 \ln (0.01)} + \frac{- 3 \ln (100)}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.2

Simplify the left side.

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Step 6.7.2.1

Cancel the common factor of $- 4 \ln (100) - 5 \ln (0.01)$ .

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Step 6.7.2.1.1

Cancel the common factor.

$\frac{x (- 4 \ln (100) - 5 \ln (0.01))}{- 4 \ln (100) - 5 \ln (0.01)} \leq \frac{- 8 \ln (0.01)}{- 4 \ln (100) - 5 \ln (0.01)} + \frac{- 3 \ln (100)}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 8 \ln (0.01)}{- 4 \ln (100) - 5 \ln (0.01)} + \frac{- 3 \ln (100)}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3

Simplify the right side.

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Step 6.7.3.1

Combine the numerators over the common denominator.

$x \leq \frac{- 8 \ln (0.01) - 3 \ln (100)}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3.2

Factor $- 1$ out of $- 8 \ln (0.01)$ .

$x \leq \frac{- (8 \ln (0.01)) - 3 \ln (100)}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3.3

Factor $- 1$ out of $- 3 \ln (100)$ .

$x \leq \frac{- (8 \ln (0.01)) - (3 \ln (100))}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3.4

Factor $- 1$ out of $- (8 \ln (0.01)) - (3 \ln (100))$ .

$x \leq \frac{- (8 \ln (0.01) + 3 \ln (100))}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3.5

Rewrite $- (8 \ln (0.01) + 3 \ln (100))$ as $- 1 (8 \ln (0.01) + 3 \ln (100))$ .

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- 4 \ln (100) - 5 \ln (0.01)}$

Step 6.7.3.6

Factor $- 1$ out of $- 4 \ln (100)$ .

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- (4 \ln (100)) - 5 \ln (0.01)}$

Step 6.7.3.7

Factor $- 1$ out of $- 5 \ln (0.01)$ .

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- (4 \ln (100)) - (5 \ln (0.01))}$

Step 6.7.3.8

Factor $- 1$ out of $- (4 \ln (100)) - (5 \ln (0.01))$ .

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- (4 \ln (100) + 5 \ln (0.01))}$

Step 6.7.3.9

Rewrite $- (4 \ln (100) + 5 \ln (0.01))$ as $- 1 (4 \ln (100) + 5 \ln (0.01))$ .

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- 1 (4 \ln (100) + 5 \ln (0.01))}$

Step 6.7.3.10

Cancel the common factor.

$x \leq \frac{- 1 (8 \ln (0.01) + 3 \ln (100))}{- 1 (4 \ln (100) + 5 \ln (0.01))}$

Step 6.7.3.11

Rewrite the expression.

$x \leq \frac{8 \ln (0.01) + 3 \ln (100)}{4 \ln (100) + 5 \ln (0.01)}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{8 \ln (0.01) + 3 \ln (100)}{4 \ln (100) + 5 \ln (0.01)}$

Interval Notation:

$(- \infty, \frac{8 \ln (0.01) + 3 \ln (100)}{4 \ln (100) + 5 \ln (0.01)}]$

Step 8