Algebra Examples

$\frac{2}{3} (1 - e^{- x}) \leq - 3$

Step 1

Simplify $\frac{2}{3} \cdot (1 - e^{- x})$ .

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Step 1.1

Apply the distributive property.

$\frac{2}{3} \cdot 1 + \frac{2}{3} (- e^{- x}) \leq - 3$

Step 1.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} + \frac{2}{3} (- e^{- x}) \leq - 3$

Step 1.3

Combine $\frac{2}{3}$ and $e^{- x}$ .

$\frac{2}{3} - \frac{2 e^{- x}}{3} \leq - 3$

Step 2

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

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Step 2.1

Subtract $\frac{2}{3}$ from both sides of the inequality.

$- \frac{2 e^{- x}}{3} \leq - 3 - \frac{2}{3}$

Step 2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 e^{- x}}{3} \leq - 3 \cdot \frac{3}{3} - \frac{2}{3}$

Step 2.3

Combine $- 3$ and $\frac{3}{3}$ .

$- \frac{2 e^{- x}}{3} \leq \frac{- 3 \cdot 3}{3} - \frac{2}{3}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{2 e^{- x}}{3} \leq \frac{- 3 \cdot 3 - 2}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 3$ by $3$ .

$- \frac{2 e^{- x}}{3} \leq \frac{- 9 - 2}{3}$

Step 2.5.2

Subtract $2$ from $- 9$ .

$- \frac{2 e^{- x}}{3} \leq \frac{- 11}{3}$

Step 2.6

Move the negative in front of the fraction.

$- \frac{2 e^{- x}}{3} \leq - \frac{11}{3}$

Step 3

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

$- 2 e^{- x} \leq - 11$

Step 4

Divide each term in $- 2 e^{- x} \leq - 11$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 e^{- x} \leq - 11$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 e^{- x}}{- 2} \geq \frac{- 11}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 e^{- x}}{- 2} \geq \frac{- 11}{- 2}$

Step 4.2.1.2

Divide $e^{- x}$ by $1$ .

$e^{- x} \geq \frac{- 11}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Dividing two negative values results in a positive value.

$e^{- x} \geq \frac{11}{2}$

Step 5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x}) \geq \ln (\frac{11}{2})$

Step 6

Expand the left side.

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Step 6.1

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

$- x \ln (e) \geq \ln (\frac{11}{2})$

Step 6.2

The natural logarithm of $e$ is $1$ .

$- x \cdot 1 \geq \ln (\frac{11}{2})$

Step 6.3

Multiply $- 1$ by $1$ .

$- x \geq \ln (\frac{11}{2})$

Step 7

Divide each term in $- x \geq \ln (\frac{11}{2})$ by $- 1$ and simplify.

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Step 7.1

Divide each term in $- x \geq \ln (\frac{11}{2})$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{\ln (\frac{11}{2})}{- 1}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{\ln (\frac{11}{2})}{- 1}$

Step 7.2.2

Divide $x$ by $1$ .

$x \leq \frac{\ln (\frac{11}{2})}{- 1}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative one from the denominator of $\frac{\ln (\frac{11}{2})}{- 1}$ .

$x \leq - 1 \cdot \ln (\frac{11}{2})$

Step 7.3.2

Rewrite $- 1 \cdot \ln (\frac{11}{2})$ as $- \ln (\frac{11}{2})$ .

$x \leq - \ln (\frac{11}{2})$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \ln (\frac{11}{2})$

Interval Notation:

$(- \infty, - \ln (\frac{11}{2})]$

Step 9