Solve for x -0.25 square root of x-6<=-3

$- 0.25 \sqrt{x} - 6 \leq - 3$

Step 1

Move all terms not containing $- 0.25 \sqrt{x}$ to the right side of the inequality.

Add $6$ to both sides of the inequality.

$- 0.25 \sqrt{x} \leq - 3 + 6$

Add $- 3$ and $6$ .

$- 0.25 \sqrt{x} \leq 3$

Step 2

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

${(- 0.25 \sqrt{x})}^{2} \leq 3^{2}$

Step 3

Simplify each side of the inequality.

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

${(- 0.25 x^{\frac{1}{2}})}^{2} \leq 3^{2}$

Simplify the left side.

Simplify ${(- 0.25 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 0.25 x^{\frac{1}{2}}$ .

${(- 0.25)}^{2} {(x^{\frac{1}{2}})}^{2} \leq 3^{2}$

Raise $- 0.25$ to the power of $2$ .

$0.0625 {(x^{\frac{1}{2}})}^{2} \leq 3^{2}$

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

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

$0.0625 x^{\frac{1}{2} \cdot 2} \leq 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$0.0625 x^{\frac{1}{2} \cdot 2} \leq 3^{2}$

Rewrite the expression.

$0.0625 x^{1} \leq 3^{2}$

Simplify.

$0.0625 x \leq 3^{2}$

Simplify the right side.

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

$0.0625 x \leq 9$

Step 4

Divide each term in $0.0625 x \leq 9$ by $0.0625$ and simplify.

Divide each term in $0.0625 x \leq 9$ by $0.0625$ .

$\frac{0.0625 x}{0.0625} \leq \frac{9}{0.0625}$

Simplify the left side.

Cancel the common factor of $0.0625$ .

Cancel the common factor.

$\frac{0.0625 x}{0.0625} \leq \frac{9}{0.0625}$

Divide $x$ by $1$ .

$x \leq \frac{9}{0.0625}$

Simplify the right side.

Divide $9$ by $0.0625$ .

$x \leq 144$

Step 5

Find the domain of $- 0.25 \sqrt{x} - 3$ .

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

$x \geq 0$

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

$[0, \infty)$

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 144$

$x > 144$

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 $x < 0$ 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.

$- 0.25 \sqrt{- 2} - 6 \leq - 3$

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

False

Test a value on the interval $0 < x < 144$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 144$ and see if this value makes the original inequality true.

$x = 2$

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

$- 0.25 \sqrt{2} - 6 \leq - 3$

The left side $- 6.35355339$ is less than the right side $- 3$ , which means that the given statement is always true.

True

Test a value on the interval $x > 144$ to see if it makes the inequality true.

Choose a value on the interval $x > 144$ and see if this value makes the original inequality true.

$x = 146$

Replace $x$ with $146$ in the original inequality.

$- 0.25 \sqrt{146} - 6 \leq - 3$

The left side $- 9.02076149$ is less than the right side $- 3$ , which means that the given statement is always true.

True

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

$x < 0$ False

$0 < x < 144$ True

$x > 144$ True

$x < 0$ False

$0 < x < 144$ True

$x > 144$ True

Step 8

The solution consists of all of the true intervals.

$0 \leq x \leq 144$ or $x \geq 144$

Step 9

Combine the intervals.

$x \geq 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \geq 0$

Interval Notation:

$[0, \infty)$

Step 11

Solve for x -0.25 square root of x-6<=-3

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