Solve the Inequality for x (4-x)^2(x-9/2)<0

${(4 - x)}^{2} (x - \frac{9}{2}) < 0$

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

${(4 - x)}^{2} = 0$

$x - \frac{9}{2} = 0$

Set ${(4 - x)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(4 - x)}^{2}$ equal to $0$ .

${(4 - x)}^{2} = 0$

Solve ${(4 - x)}^{2} = 0$ for $x$ .

Set the $4 - x$ equal to $0$ .

$4 - x = 0$

Solve for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

Set $x - \frac{9}{2}$ equal to $0$ and solve for $x$ .

Set $x - \frac{9}{2}$ equal to $0$ .

$x - \frac{9}{2} = 0$

Add $\frac{9}{2}$ to both sides of the equation.

$x = \frac{9}{2}$

The final solution is all the values that make ${(4 - x)}^{2} (x - \frac{9}{2}) < 0$ true.

$x = 4, \frac{9}{2}$

Use each root to create test intervals.

$x < 4$

$4 < x < \frac{9}{2}$

$x > \frac{9}{2}$

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 $x < 4$ to see if it makes the inequality true.

Choose a value on the interval $x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

${(4 - (0))}^{2} ((0) - \frac{9}{2}) < 0$

The left side $- 72$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $4 < x < \frac{9}{2}$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < \frac{9}{2}$ and see if this value makes the original inequality true.

$x = 4.25$

Replace $x$ with $4.25$ in the original inequality.

${(4 - (4.25))}^{2} ((4.25) - \frac{9}{2}) < 0$

The left side $- 0.015625$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{9}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{9}{2}$ and see if this value makes the original inequality true.

$x = 7$

Replace $x$ with $7$ in the original inequality.

${(4 - (7))}^{2} ((7) - \frac{9}{2}) < 0$

The left side $22.5$ 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.

$x < 4$ True

$4 < x < \frac{9}{2}$ True

$x > \frac{9}{2}$ False

$x < 4$ True

$4 < x < \frac{9}{2}$ True

$x > \frac{9}{2}$ False

The solution consists of all of the true intervals.

$x < 4$ or $4 < x < \frac{9}{2}$

The result can be shown in multiple forms.

Inequality Form:

$x < 4 or 4 < x < \frac{9}{2}$

Interval Notation:

$(- \infty, 4) \cup (4, \frac{9}{2})$

Solve the Inequality for x (4-x)^2(x-9/2)<0

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