Solve the Inequality for x square root of 4x-4-2<4

$\sqrt{4 x - 4} - 2 < 4$

Move all terms not containing $\sqrt{4 x - 4}$ to the right side of the inequality.

Add $2$ to both sides of the inequality.

$\sqrt{4 x - 4} < 4 + 2$

Add $4$ and $2$ .

$\sqrt{4 x - 4} < 6$

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

${\sqrt{4 x - 4}}^{2} < 6^{2}$

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 4}$ as ${(4 x - 4)}^{\frac{1}{2}}$ .

${({(4 x - 4)}^{\frac{1}{2}})}^{2} < 6^{2}$

Simplify the left side.

Simplify ${({(4 x - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x - 4)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x - 4)}^{\frac{1}{2} \cdot 2} < 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x - 4)}^{\frac{1}{2} \cdot 2} < 6^{2}$

Rewrite the expression.

${(4 x - 4)}^{1} < 6^{2}$

Simplify.

$4 x - 4 < 6^{2}$

Simplify the right side.

Raise $6$ to the power of $2$ .

$4 x - 4 < 36$

Solve for $x$ .

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

Add $4$ to both sides of the inequality.

$4 x < 36 + 4$

Add $36$ and $4$ .

$4 x < 40$

Divide each term in $4 x < 40$ by $4$ and simplify.

Divide each term in $4 x < 40$ by $4$ .

$\frac{4 x}{4} < \frac{40}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{40}{4}$

Divide $x$ by $1$ .

$x < \frac{40}{4}$

Simplify the right side.

Divide $40$ by $4$ .

$x < 10$

Find the domain of $2 \sqrt{x - 1} - 6$ .

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

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

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

$[1, \infty)$

Use each root to create test intervals.

$x < 1$

$1 < x < 10$

$x > 10$

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 $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$\sqrt{4 (0) - 4} - 2 < 4$

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 $1 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 10$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$\sqrt{4 (6) - 4} - 2 < 4$

The left side $2.47213595$ is less than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\sqrt{4 (12) - 4} - 2 < 4$

The left side $4.63324958$ is not less than the right side $4$ , which means that the given statement is false.

False

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

$x < 1$ False

$1 < x < 10$ True

$x > 10$ False

$x < 1$ False

$1 < x < 10$ True

$x > 10$ False

The solution consists of all of the true intervals.

$1 < x < 10$

The result can be shown in multiple forms.

Inequality Form:

$1 < x < 10$

Interval Notation:

$(1, 10)$

Solve the Inequality for x square root of 4x-4-2<4

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