Solve the Inequality for x x(2x+1)(5x-30)<=0

$x (2 x + 1) (5 x - 30) \leq 0$

Step 1

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

$x = 0$

$2 x + 1 = 0$

$5 x - 30 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 4

Set $5 x - 30$ equal to $0$ and solve for $x$ .

Set $5 x - 30$ equal to $0$ .

$5 x - 30 = 0$

Solve $5 x - 30 = 0$ for $x$ .

Add $30$ to both sides of the equation.

$5 x = 30$

Divide each term in $5 x = 30$ by $5$ and simplify.

Divide each term in $5 x = 30$ by $5$ .

$\frac{5 x}{5} = \frac{30}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{30}{5}$

Divide $x$ by $1$ .

$x = \frac{30}{5}$

Simplify the right side.

Divide $30$ by $5$ .

$x = 6$

Step 5

The final solution is all the values that make $x (2 x + 1) (5 x - 30) \leq 0$ true.

$x = 0, - \frac{1}{2}, 6$

Step 6

Use each root to create test intervals.

$x < - \frac{1}{2}$

$- \frac{1}{2} < x < 0$

$0 < x < 6$

$x > 6$

Step 7

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 $x < - \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{1}{2}$ and see if this value makes the original inequality true.

$x = - 3$

Replace $x$ with $- 3$ in the original inequality.

$(- 3) (2 (- 3) + 1) (5 (- 3) - 30) \leq 0$

The left side $- 675$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{1}{2} < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{1}{2} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.25$

Replace $x$ with $- 0.25$ in the original inequality.

$(- 0.25) (2 (- 0.25) + 1) (5 (- 0.25) - 30) \leq 0$

The left side $3.90625$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 6$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$(3) (2 (3) + 1) (5 (3) - 30) \leq 0$

The left side $- 315$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$(8) (2 (8) + 1) (5 (8) - 30) \leq 0$

The left side $1360$ is greater 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.

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < 0$ False

$0 < x < 6$ True

$x > 6$ False

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < 0$ False

$0 < x < 6$ True

$x > 6$ False

Step 8

The solution consists of all of the true intervals.

$x \leq - \frac{1}{2}$ or $0 \leq x \leq 6$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{1}{2} or 0 \leq x \leq 6$

Interval Notation:

$(- \infty, - \frac{1}{2}] \cup [0, 6]$

Step 10

Solve the Inequality for x x(2x+1)(5x-30)<=0

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