Finite Math Examples

$\log_{2} (182) - 2 \log_{2} (\sqrt{5 - x}) = \log_{2} (11 - x) + 1$

Step 1

Set the argument in $\log_{2} (\sqrt{5 - x})$ greater than $0$ to find where the expression is defined.

$\sqrt{5 - x} > 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

${\sqrt{5 - x}}^{2} > 0^{2}$

Step 2.2

Simplify each side of the inequality.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - x}$ as ${(5 - x)}^{\frac{1}{2}}$ .

${({(5 - x)}^{\frac{1}{2}})}^{2} > 0^{2}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Simplify ${({(5 - x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.1

Multiply the exponents in ${({(5 - x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.1.1

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

${(5 - x)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1.2.1

Cancel the common factor.

${(5 - x)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(5 - x)}^{1} > 0^{2}$

Step 2.2.2.1.2

Simplify.

$5 - x > 0^{2}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$5 - x > 0$

Step 2.3

Solve for $x$ .

Tap for more steps...

Step 2.3.1

Subtract $5$ from both sides of the inequality.

$- x > - 5$

Step 2.3.2

Divide each term in $- x > - 5$ by $- 1$ and simplify.

Tap for more steps...

Step 2.3.2.1

Divide each term in $- x > - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 5}{- 1}$

Step 2.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 5}{- 1}$

Step 2.3.2.2.2

Divide $x$ by $1$ .

$x < \frac{- 5}{- 1}$

Step 2.3.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.2.3.1

Divide $- 5$ by $- 1$ .

$x < 5$

Step 2.4

Find the domain of $\sqrt{5 - x}$ .

Tap for more steps...

Step 2.4.1

Set the radicand in $\sqrt{5 - x}$ greater than or equal to $0$ to find where the expression is defined.

$5 - x \geq 0$

Step 2.4.2

Solve for $x$ .

Tap for more steps...

Step 2.4.2.1

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Step 2.4.2.2

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

Tap for more steps...

Step 2.4.2.2.1

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 5}{- 1}$

Step 2.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 5}{- 1}$

Step 2.4.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 1}$

Step 2.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.4.2.2.3.1

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 2.4.3

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

$(- \infty, 5]$

Step 2.5

The solution consists of all of the true intervals.

$x < 5$

Step 3

Set the radicand in $\sqrt{5 - x}$ greater than or equal to $0$ to find where the expression is defined.

$5 - x \geq 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Step 4.2

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 5}{- 1}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 5}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 1}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 5

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

Interval Notation:

$(- \infty, 5)$

Set-Builder Notation:

${x | x < 5}$

Step 6