Solve for y |y-5|>3

$| y - 5 | > 3$

Step 1

Write $| y - 5 | > 3$ as a piecewise.

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

$y - 5 \geq 0$

Add $5$ to both sides of the inequality.

$y \geq 5$

In the piece where $y - 5$ is non-negative, remove the absolute value.

$y - 5 > 3$

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

$y - 5 < 0$

Add $5$ to both sides of the inequality.

$y < 5$

In the piece where $y - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y - 5) > 3$

Write as a piecewise.

${\begin{cases} y - 5 > 3 & y \geq 5 \\ - (y - 5) > 3 & y < 5 \end{cases}$

Simplify $- (y - 5) > 3$ .

Apply the distributive property.

${\begin{cases} y - 5 > 3 & y \geq 5 \\ - y - - 5 > 3 & y < 5 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} y - 5 > 3 & y \geq 5 \\ - y + 5 > 3 & y < 5 \end{cases}$

Step 2

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

Add $5$ to both sides of the inequality.

$y > 3 + 5$

Add $3$ and $5$ .

$y > 8$

Step 3

Solve $- y + 5 > 3$ for $y$ .

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

Subtract $5$ from both sides of the inequality.

$- y > 3 - 5$

Subtract $5$ from $3$ .

$- y > - 2$

Divide each term in $- y > - 2$ by $- 1$ and simplify.

Divide each term in $- y > - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{- 2}{- 1}$

Divide $y$ by $1$ .

$y < \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$y < 2$

Step 4

Find the union of the solutions.

$y < 2$ or $y > 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$y < 2 or y > 8$

Interval Notation:

$(- \infty, 2) \cup (8, \infty)$

Step 6

Solve for y |y-5|>3

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