Algebra Examples

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

Step 1

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

$x = - 5, 5$

Step 2

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 5$

$x > 5$

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

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Step 3.1.1

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

$x = - 8$

Step 3.1.2

Replace $x$ with $- 8$ in the original inequality.

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

Step 3.1.3

The left side $9.984 E - 5$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 3.2

Test a value on the interval $- 5 < x < 5$ to see if it makes the inequality true.

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Step 3.2.1

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

$x = 0$

Step 3.2.2

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

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

Step 3.2.3

The left side $- 25$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 3.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 3.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 3.3.2

Replace $x$ with $8$ in the original inequality.

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

Step 3.3.3

The left side $15234375$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 3.4

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

$x < - 5$ False

$- 5 < x < 5$ True

$x > 5$ False

$x < - 5$ False

$- 5 < x < 5$ True

$x > 5$ False

Step 4

The solution consists of all of the true intervals.

$- 5 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < 5$

Interval Notation:

$(- 5, 5)$

Step 6