Solve the Inequality for b square root of b-7+6<=12

$\sqrt{b - 7} + 6 \leq 12$

Move all terms not containing $\sqrt{b - 7}$ to the right side of the inequality.

Subtract $6$ from both sides of the inequality.

$\sqrt{b - 7} \leq 12 - 6$

Subtract $6$ from $12$ .

$\sqrt{b - 7} \leq 6$

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

${\sqrt{b - 7}}^{2} \leq 6^{2}$

Simplify each side of the inequality.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b - 7}$ as ${(b - 7)}^{\frac{1}{2}}$ .

${({(b - 7)}^{\frac{1}{2}})}^{2} \leq 6^{2}$

Simplify the left side.

Simplify ${({(b - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(b - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(b - 7)}^{\frac{1}{2} \cdot 2} \leq 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(b - 7)}^{\frac{1}{2} \cdot 2} \leq 6^{2}$

Rewrite the expression.

${(b - 7)}^{1} \leq 6^{2}$

Simplify.

$b - 7 \leq 6^{2}$

Simplify the right side.

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

$b - 7 \leq 36$

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

Add $7$ to both sides of the inequality.

$b \leq 36 + 7$

Add $36$ and $7$ .

$b \leq 43$

Find the domain of $\sqrt{b - 7} - 6$ .

Set the radicand in $\sqrt{b - 7}$ greater than or equal to $0$ to find where the expression is defined.

$b - 7 \geq 0$

Add $7$ to both sides of the inequality.

$b \geq 7$

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

$[7, \infty)$

Use each root to create test intervals.

$b < 7$

$7 < b < 43$

$b > 43$

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 $b < 7$ to see if it makes the inequality true.

Choose a value on the interval $b < 7$ and see if this value makes the original inequality true.

$b = 0$

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

$\sqrt{(0) - 7} + 6 \leq 12$

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 $7 < b < 43$ to see if it makes the inequality true.

Choose a value on the interval $7 < b < 43$ and see if this value makes the original inequality true.

$b = 10$

Replace $b$ with $10$ in the original inequality.

$\sqrt{(10) - 7} + 6 \leq 12$

The left side $7.7320508$ is less than the right side $12$ , which means that the given statement is always true.

True

Test a value on the interval $b > 43$ to see if it makes the inequality true.

Choose a value on the interval $b > 43$ and see if this value makes the original inequality true.

$b = 46$

Replace $b$ with $46$ in the original inequality.

$\sqrt{(46) - 7} + 6 \leq 12$

The left side $12.24499799$ is greater than the right side $12$ , which means that the given statement is false.

False

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

$b < 7$ False

$7 < b < 43$ True

$b > 43$ False

$b < 7$ False

$7 < b < 43$ True

$b > 43$ False

The solution consists of all of the true intervals.

$7 \leq b \leq 43$

The result can be shown in multiple forms.

Inequality Form:

$7 \leq b \leq 43$

Interval Notation:

$[7, 43]$

Solve the Inequality for b square root of b-7+6<=12

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