Solve the Inequality for x x^2*5^x-5^(2+x)<0

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

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

$x = - 5, 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 5$

$x > 5$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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

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

$x = - 8$

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

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

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

False

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

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

$x = 0$

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

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

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

True

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

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

$x = 8$

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

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

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

False

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

The solution consists of all of the true intervals.

$- 5 < x < 5$

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < 5$

Interval Notation:

$(- 5, 5)$