Solve the Inequality for y 2+ square root of 4y-4<=6

$2 + \sqrt{4 y - 4} \leq 6$

Step 1

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

Subtract $2$ from both sides of the inequality.

$\sqrt{4 y - 4} \leq 6 - 2$

Subtract $2$ from $6$ .

$\sqrt{4 y - 4} \leq 4$

Step 2

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

${\sqrt{4 y - 4}}^{2} \leq 4^{2}$

Step 3

Simplify each side of the inequality.

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

${({(4 y - 4)}^{\frac{1}{2}})}^{2} \leq 4^{2}$

Simplify the left side.

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

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

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

${(4 y - 4)}^{\frac{1}{2} \cdot 2} \leq 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 y - 4)}^{\frac{1}{2} \cdot 2} \leq 4^{2}$

Rewrite the expression.

${(4 y - 4)}^{1} \leq 4^{2}$

Simplify.

$4 y - 4 \leq 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$4 y - 4 \leq 16$

Step 4

Solve for $y$ .

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

Add $4$ to both sides of the inequality.

$4 y \leq 16 + 4$

Add $16$ and $4$ .

$4 y \leq 20$

Divide each term in $4 y \leq 20$ by $4$ and simplify.

Divide each term in $4 y \leq 20$ by $4$ .

$\frac{4 y}{4} \leq \frac{20}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} \leq \frac{20}{4}$

Divide $y$ by $1$ .

$y \leq \frac{20}{4}$

Simplify the right side.

Divide $20$ by $4$ .

$y \leq 5$

Step 5

Find the domain of $- 4 + 2 \sqrt{y - 1}$ .

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

$y - 1 \geq 0$

Add $1$ to both sides of the inequality.

$y \geq 1$

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

$[1, \infty)$

Step 6

Use each root to create test intervals.

$y < 1$

$1 < y < 5$

$y > 5$

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

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

$y = 0$

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

$2 + \sqrt{4 (0) - 4} \leq 6$

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 $1 < y < 5$ to see if it makes the inequality true.

Choose a value on the interval $1 < y < 5$ and see if this value makes the original inequality true.

$y = 3$

Replace $y$ with $3$ in the original inequality.

$2 + \sqrt{4 (3) - 4} \leq 6$

The left side $4.82842712$ is less than the right side $6$ , which means that the given statement is always true.

True

Test a value on the interval $y > 5$ to see if it makes the inequality true.

Choose a value on the interval $y > 5$ and see if this value makes the original inequality true.

$y = 8$

Replace $y$ with $8$ in the original inequality.

$2 + \sqrt{4 (8) - 4} \leq 6$

The left side $7.29150262$ is greater than the right side $6$ , which means that the given statement is false.

False

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

$y < 1$ False

$1 < y < 5$ True

$y > 5$ False

$y < 1$ False

$1 < y < 5$ True

$y > 5$ False

Step 8

The solution consists of all of the true intervals.

$1 \leq y \leq 5$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$1 \leq y \leq 5$

Interval Notation:

$[1, 5]$

Step 10

Solve the Inequality for y 2+ square root of 4y-4<=6

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