Solve for x square root of 1-x>3

$\sqrt{1 - x} > 3$

Step 1

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

${\sqrt{1 - x}}^{2} > 3^{2}$

Step 2

Simplify each side of the inequality.

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

${({(1 - x)}^{\frac{1}{2}})}^{2} > 3^{2}$

Simplify the left side.

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

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

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

${(1 - x)}^{\frac{1}{2} \cdot 2} > 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(1 - x)}^{\frac{1}{2} \cdot 2} > 3^{2}$

Rewrite the expression.

${(1 - x)}^{1} > 3^{2}$

Simplify.

$1 - x > 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$1 - x > 9$

Step 3

Solve for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $1$ from both sides of the inequality.

$- x > 9 - 1$

Subtract $1$ from $9$ .

$- x > 8$

Divide each term in $- x > 8$ by $- 1$ and simplify.

Divide each term in $- x > 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{8}{- 1}$

Divide $x$ by $1$ .

$x < \frac{8}{- 1}$

Simplify the right side.

Divide $8$ by $- 1$ .

$x < - 8$

Step 4

Find the domain of $\sqrt{1 - x} - 3$ .

Set the radicand in $\sqrt{1 - x}$ greater than or equal to $0$ to find where the expression is defined.

$1 - x \geq 0$

Solve for $x$ .

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x \leq 1$

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

$(- \infty, 1]$

Step 5

The solution consists of all of the true intervals.

$x < - 8$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - 8$

Interval Notation:

$(- \infty, - 8)$

Step 7

Solve for x square root of 1-x>3

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