Solve the Inequality for x square root of 6-x<x

$\sqrt{6 - x} < x$

Step 1

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

${\sqrt{6 - x}}^{2} < x^{2}$

Step 2

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 - x}$ as ${(6 - x)}^{\frac{1}{2}}$ .

${({(6 - x)}^{\frac{1}{2}})}^{2} < x^{2}$

Simplify the left side.

Simplify ${({(6 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(6 - x)}^{\frac{1}{2} \cdot 2} < x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 - x)}^{\frac{1}{2} \cdot 2} < x^{2}$

Rewrite the expression.

${(6 - x)}^{1} < x^{2}$

Simplify.

$6 - x < x^{2}$

Step 3

Solve for $x$ .

Subtract $x^{2}$ from both sides of the inequality.

$6 - x - x^{2} < 0$

Convert the inequality to an equation.

$6 - x - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $6 - x - x^{2}$ .

Reorder the expression.

Move $6$ .

$- x - x^{2} + 6 = 0$

Reorder $- x$ and $- x^{2}$ .

$- x^{2} - x + 6 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - x + 6 = 0$

Factor $- 1$ out of $- x$ .

$- (x^{2}) - (x) + 6 = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$- (x^{2}) - (x) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (x^{2}) - (x)$ .

$- (x^{2} + x) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 6)$ .

$- (x^{2} + x - 6) = 0$

Factor.

Factor $x^{2} + x - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Write the factored form using these integers.

$- ((x - 2) (x + 3)) = 0$

Remove unnecessary parentheses.

$- (x - 2) (x + 3) = 0$

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

$x - 2 = 0$

$x + 3 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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.

$x = - 3$

The final solution is all the values that make $- (x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

Step 4

Find the domain of $\sqrt{6 - x} - x$ .

Set the radicand in $\sqrt{6 - x}$ greater than or equal to $0$ to find where the expression is defined.

$6 - x \geq 0$

Solve for $x$ .

Subtract $6$ from both sides of the inequality.

$- x \geq - 6$

Divide each term in $- x \geq - 6$ by $- 1$ and simplify.

Divide each term in $- x \geq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x \leq 6$

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

$(- \infty, 6]$

Step 5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 2$

$2 < x < 6$

$x > 6$

Step 6

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.

$\sqrt{6 - (- 6)} < - 6$

The left side $3.46410161$ is not less than the right side $- 6$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\sqrt{6 - (0)} < 0$

The left side $2.44948974$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $2 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 6$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$\sqrt{6 - (4)} < 4$

The left side $1.41421356$ is less than the right side $4$ , 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.

$\sqrt{6 - (8)} < 8$

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.

$x < - 3$ False

$- 3 < x < 2$ False

$2 < x < 6$ True

$x > 6$ False

$x < - 3$ False

$- 3 < x < 2$ False

$2 < x < 6$ True

$x > 6$ False

Step 7

The solution consists of all of the true intervals.

$2 < x < 6$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 6$

Interval Notation:

$(2, 6)$

Step 9

Solve the Inequality for x square root of 6-x<x

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