Algebra Examples

x^{2} \cdot 5^{x} - 5^{2 + x} < 0

$x^{2} \cdot 5^{x} - 5^{2 + x} < 0$

Step 1

Graph each side of the equation. The solution is the x-value of the point of intersection.

x = - 5, 5

$x = - 5, 5$

Step 2

Use each root to create test intervals.

x < - 5

$x < - 5$

- 5 < x < 5

$- 5 < x < 5$

x > 5

$x > 5$

Step 3

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 3.1

Test a value on the interval

x < - 5

$x < - 5$ to see if it makes the inequality true.

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Step 3.1.1

Choose a value on the interval

x < - 5

$x < - 5$ and see if this value makes the original inequality true.

x = - 8

$x = - 8$

Step 3.1.2

Replace

x

$x$ with

- 8

$- 8$ in the original inequality.

{(- 8)}^{2} \cdot 5^{- 8} - 5^{2 - 8} < 0

${(- 8)}^{2} \cdot 5^{- 8} - 5^{2 - 8} < 0$

Step 3.1.3

The left side

9.984 E - 5

$9.984 E - 5$ is not less than the right side

0

$0$ , which means that the given statement is false.

False

Step 3.2

Test a value on the interval

- 5 < x < 5

$- 5 < x < 5$ to see if it makes the inequality true.

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Step 3.2.1

Choose a value on the interval

- 5 < x < 5

$- 5 < x < 5$ and see if this value makes the original inequality true.

x = 0

$x = 0$

Step 3.2.2

Replace

x

$x$ with

0

$0$ in the original inequality.

{(0)}^{2} \cdot 5^{0} - 5^{2 + 0} < 0

${(0)}^{2} \cdot 5^{0} - 5^{2 + 0} < 0$

Step 3.2.3

The left side

- 25

$- 25$ is less than the right side

0

$0$ , which means that the given statement is always true.

True

Step 3.3

Test a value on the interval

x > 5

$x > 5$ to see if it makes the inequality true.

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Step 3.3.1

Choose a value on the interval

x > 5

$x > 5$ and see if this value makes the original inequality true.

x = 8

$x = 8$

Step 3.3.2

Replace

x

$x$ with

8

$8$ in the original inequality.

{(8)}^{2} \cdot 5^{8} - 5^{2 + 8} < 0

${(8)}^{2} \cdot 5^{8} - 5^{2 + 8} < 0$

Step 3.3.3

The left side

15234375

$15234375$ is not less than the right side

0

$0$ , which means that the given statement is false.

False

Step 3.4

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

x < - 5

$x < - 5$ False

- 5 < x < 5

$- 5 < x < 5$ True

x > 5

$x > 5$ False

x < - 5

$x < - 5$ False

- 5 < x < 5

$- 5 < x < 5$ True

x > 5

$x > 5$ False

Step 4

The solution consists of all of the true intervals.

- 5 < x < 5

$- 5 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

- 5 < x < 5

$- 5 < x < 5$

Interval Notation:

(- 5, 5)

$(- 5, 5)$

Step 6