Solve the Inequality for x square root of -x<0

$\sqrt{- x} < 0$

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{- x}}^{2} < 0^{2}$

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x}$ as ${(- x)}^{\frac{1}{2}}$ .

${({(- x)}^{\frac{1}{2}})}^{2} < 0^{2}$

Simplify the left side.

Simplify ${({(- x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- x)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- x)}^{\frac{1}{2} \cdot 2} < 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- x)}^{\frac{1}{2} \cdot 2} < 0^{2}$

Rewrite the expression.

${(- x)}^{1} < 0^{2}$

Simplify.

$- x < 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$- x < 0$

Divide each term in $- x < 0$ by $- 1$ and simplify.

Divide each term in $- x < 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{0}{- 1}$

Divide $x$ by $1$ .

$x > \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x > 0$

Find the domain of $\sqrt{- x}$ .

Set the radicand in $\sqrt{- x}$ greater than or equal to $0$ to find where the expression is defined.

$- x \geq 0$

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x \leq 0$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0]$

Use each root to create test intervals.

$x < 0$

$x > 0$

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 $x < 0$ to see if it makes the inequality true.

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

$x = - 2$

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

$\sqrt{- (- 2)} < 0$

The left side $1.41421356$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 2$

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

$\sqrt{- (2)} < 0$

The left side is not equal to the right side, which means that the given statement is false.

False

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

$x < 0$ False

$x > 0$ False

$x < 0$ False

$x > 0$ False

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Solve the Inequality for x square root of -x<0

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