Solve the Inequality for x x(x+3)(5-x)>=0

$x (x + 3) (5 - x) \geq 0$

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

$x = 0$

$x + 3 = 0$

$5 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $5 - x$ equal to $0$ and solve for $x$ .

Set $5 - x$ equal to $0$ .

$5 - x = 0$

Solve $5 - x = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$- x = - 5$

Divide each term in $- x = - 5$ by $- 1$ and simplify.

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x = 5$

The final solution is all the values that make $x (x + 3) (5 - x) \geq 0$ true.

$x = 0, - 3, 5$

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 0$

$0 < x < 5$

$x > 5$

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 $x < - 3$ to see if it makes the inequality true.

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

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$(- 6) ((- 6) + 3) (5 - (- 6)) \geq 0$

The left side $198$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 3 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$(- 2) ((- 2) + 3) (5 - (- 2)) \geq 0$

The left side $- 14$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 5$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$(2) ((2) + 3) (5 - (2)) \geq 0$

The left side $30$ is greater 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.

$(8) ((8) + 3) (5 - (8)) \geq 0$

The left side $- 264$ is 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.

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 5$ True

$x > 5$ False

$x < - 3$ True

$- 3 < x < 0$ False

$0 < x < 5$ True

$x > 5$ False

The solution consists of all of the true intervals.

$x \leq - 3$ or $0 \leq x \leq 5$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or 0 \leq x \leq 5$

Interval Notation:

$(- \infty, - 3] \cup [0, 5]$

Solve the Inequality for x x(x+3)(5-x)>=0

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