Precalculus Examples

$\log_{3} ({(1 + y)}^{2}) = 2$

Step 1

Set the argument in $\log_{3} ({(1 + y)}^{2})$ greater than $0$ to find where the expression is defined.

${(1 + y)}^{2} > 0$

Step 2

Solve for $y$ .

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Step 2.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(1 + y)}^{2}} > \sqrt{0}$

Step 2.2

Simplify the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Pull terms out from under the radical.

$| 1 + y | > \sqrt{0}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\sqrt{0}$ .

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Step 2.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| 1 + y | > \sqrt{0^{2}}$

Step 2.2.2.1.2

Pull terms out from under the radical.

$| 1 + y | > | 0 |$

Step 2.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| 1 + y | > 0$

Step 2.3

Write $| 1 + y | > 0$ as a piecewise.

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Step 2.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$1 + y \geq 0$

Step 2.3.2

Subtract $1$ from both sides of the inequality.

$y \geq - 1$

Step 2.3.3

In the piece where $1 + y$ is non-negative, remove the absolute value.

$1 + y > 0$

Step 2.3.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$1 + y < 0$

Step 2.3.5

Subtract $1$ from both sides of the inequality.

$y < - 1$

Step 2.3.6

In the piece where $1 + y$ is negative, remove the absolute value and multiply by $- 1$ .

$- (1 + y) > 0$

Step 2.3.7

Write as a piecewise.

${\begin{cases} 1 + y > 0 & y \geq - 1 \\ - (1 + y) > 0 & y < - 1 \end{cases}$

Step 2.3.8

Simplify $- (1 + y) > 0$ .

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Step 2.3.8.1

Apply the distributive property.

${\begin{cases} 1 + y > 0 & y \geq - 1 \\ - 1 \cdot 1 - y > 0 & y < - 1 \end{cases}$

Step 2.3.8.2

Multiply $- 1$ by $1$ .

${\begin{cases} 1 + y > 0 & y \geq - 1 \\ - 1 - y > 0 & y < - 1 \end{cases}$

Step 2.4

Subtract $1$ from both sides of the inequality.

$y > - 1$

Step 2.5

Solve $- 1 - y > 0$ for $y$ .

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Step 2.5.1

Add $1$ to both sides of the inequality.

$- y > 1$

Step 2.5.2

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

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Step 2.5.2.1

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

$\frac{- y}{- 1} < \frac{1}{- 1}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{1}{- 1}$

Step 2.5.2.2.2

Divide $y$ by $1$ .

$y < \frac{1}{- 1}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $1$ by $- 1$ .

$y < - 1$

Step 2.6

Find the union of the solutions.

$y < - 1$ or $y > - 1$

Step 3

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, \infty)$

Set-Builder Notation:

${y | y \neq - 1}$

Step 4