Precalculus Examples

$x^{2} e^{x} - 3 e^{x} < 0$

Step 1

Convert the inequality to an equation.

$x^{2} e^{x} - 3 e^{x} = 0$

Step 2

Factor $e^{x}$ out of $x^{2} e^{x} - 3 e^{x}$ .

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Step 2.1

Factor $e^{x}$ out of $x^{2} e^{x}$ .

$e^{x} x^{2} - 3 e^{x} = 0$

Step 2.2

Factor $e^{x}$ out of $- 3 e^{x}$ .

$e^{x} x^{2} + e^{x} \cdot - 3 = 0$

Step 2.3

Factor $e^{x}$ out of $e^{x} x^{2} + e^{x} \cdot - 3$ .

$e^{x} (x^{2} - 3) = 0$

Step 3

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

$e^{x} = 0$

$x^{2} - 3 = 0$

Step 4

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 4.2

Solve $e^{x} = 0$ for $x$ .

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Step 4.2.1

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

$\ln (e^{x}) = \ln (0)$

Step 4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 4.2.3

There is no solution for $e^{x} = 0$

No solution

Step 5

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 5.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 5.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3}$

Step 5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{3}$

Step 5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{3}$

Step 5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{3}, - \sqrt{3}$

Step 6

The final solution is all the values that make $e^{x} (x^{2} - 3) = 0$ true.

$x = \sqrt{3}, - \sqrt{3}$

Step 7

Use each root to create test intervals.

$x < - \sqrt{3}$

$- \sqrt{3} < x < \sqrt{3}$

$x > \sqrt{3}$

Step 8

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 8.1

Test a value on the interval $x < - \sqrt{3}$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < - \sqrt{3}$ and see if this value makes the original inequality true.

$x = - 4$

Step 8.1.2

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

${(- 4)}^{2} e^{- 4} - 3 e^{- 4} < 0$

Step 8.1.3

The left side $0.2381033$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $- \sqrt{3} < x < \sqrt{3}$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $- \sqrt{3} < x < \sqrt{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 8.2.2

Replace $x$ with $0$ in the original inequality.

${(0)}^{2} e^{0} - 3 e^{0} < 0$

Step 8.2.3

The left side $- 3$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > \sqrt{3}$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > \sqrt{3}$ and see if this value makes the original inequality true.

$x = 4$

Step 8.3.2

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

${(4)}^{2} e^{4} - 3 e^{4} < 0$

Step 8.3.3

The left side $709.77595043$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - \sqrt{3}$ False

$- \sqrt{3} < x < \sqrt{3}$ True

$x > \sqrt{3}$ False

$x < - \sqrt{3}$ False

$- \sqrt{3} < x < \sqrt{3}$ True

$x > \sqrt{3}$ False

Step 9

The solution consists of all of the true intervals.

$- \sqrt{3} < x < \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- \sqrt{3} < x < \sqrt{3}$

Interval Notation:

$(- \sqrt{3}, \sqrt{3})$

Step 11