Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $6$ from both sides of the inequality.

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

Step 1.2

Subtract $6$ from $12$ .

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

Step 2

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

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

Step 3

Simplify each side of the inequality.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$b - 7 \leq 36$

Step 4

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

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Step 4.1

Add $7$ to both sides of the inequality.

$b \leq 36 + 7$

Step 4.2

Add $36$ and $7$ .

$b \leq 43$

Step 5

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

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Step 5.1

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

$b - 7 \geq 0$

Step 5.2

Add $7$ to both sides of the inequality.

$b \geq 7$

Step 5.3

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

$[7, \infty)$

Step 6

Use each root to create test intervals.

$b < 7$

$7 < b < 43$

$b > 43$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $b < 7$ to see if it makes the inequality true.

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Step 7.1.1

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

$b = 0$

Step 7.1.2

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

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

Step 7.1.3

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

False

Step 7.2

Test a value on the interval $7 < b < 43$ to see if it makes the inequality true.

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Step 7.2.1

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

$b = 10$

Step 7.2.2

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

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

Step 7.2.3

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

True

Step 7.3

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

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Step 7.3.1

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

$b = 46$

Step 7.3.2

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

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

Step 7.3.3

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

False

Step 7.4

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

Step 8

The solution consists of all of the true intervals.

$7 \leq b \leq 43$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$7 \leq b \leq 43$

Interval Notation:

$[7, 43]$

Step 10