Algebra Examples

$| y + 2 | > 6$

Step 1

Write $| y + 2 | > 6$ as a piecewise.

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Step 1.1

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

$y + 2 \geq 0$

Step 1.2

Subtract $2$ from both sides of the inequality.

$y \geq - 2$

Step 1.3

In the piece where $y + 2$ is non-negative, remove the absolute value.

$y + 2 > 6$

Step 1.4

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

$y + 2 < 0$

Step 1.5

Subtract $2$ from both sides of the inequality.

$y < - 2$

Step 1.6

In the piece where $y + 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y + 2) > 6$

Step 1.7

Write as a piecewise.

${\begin{cases} y + 2 > 6 & y \geq - 2 \\ - (y + 2) > 6 & y < - 2 \end{cases}$

Step 1.8

Simplify $- (y + 2) > 6$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} y + 2 > 6 & y \geq - 2 \\ - y - 1 \cdot 2 > 6 & y < - 2 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $2$ .

${\begin{cases} y + 2 > 6 & y \geq - 2 \\ - y - 2 > 6 & y < - 2 \end{cases}$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the inequality.

$y > 6 - 2$

Step 2.2

Subtract $2$ from $6$ .

$y > 4$

Step 3

Solve $- y - 2 > 6$ for $y$ .

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Step 3.1

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

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Step 3.1.1

Add $2$ to both sides of the inequality.

$- y > 6 + 2$

Step 3.1.2

Add $6$ and $2$ .

$- y > 8$

Step 3.2

Divide each term in $- y > 8$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y > 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{8}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{8}{- 1}$

Step 3.2.2.2

Divide $y$ by $1$ .

$y < \frac{8}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $8$ by $- 1$ .

$y < - 8$

Step 4

Find the union of the solutions.

$y < - 8$ or $y > 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$y < - 8 or y > 4$

Interval Notation:

$(- \infty, - 8) \cup (4, \infty)$

Step 6