Solve for y |y-15|<23

$| y - 15 | < 23$

Write $| y - 15 | < 23$ as a piecewise.

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

$y - 15 \geq 0$

Add $15$ to both sides of the inequality.

$y \geq 15$

In the piece where $y - 15$ is non-negative, remove the absolute value.

$y - 15 < 23$

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

$y - 15 < 0$

Add $15$ to both sides of the inequality.

$y < 15$

In the piece where $y - 15$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y - 15) < 23$

Write as a piecewise.

${\begin{cases} y - 15 < 23 & y \geq 15 \\ - (y - 15) < 23 & y < 15 \end{cases}$

Simplify $- (y - 15) < 23$ .

Apply the distributive property.

${\begin{cases} y - 15 < 23 & y \geq 15 \\ - y - - 15 < 23 & y < 15 \end{cases}$

Multiply $- 1$ by $- 15$ .

${\begin{cases} y - 15 < 23 & y \geq 15 \\ - y + 15 < 23 & y < 15 \end{cases}$

Solve $y - 15 < 23$ when $y \geq 15$ .

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

Add $15$ to both sides of the inequality.

$y < 23 + 15$

Add $23$ and $15$ .

$y < 38$

Find the intersection of $y < 38$ and $y \geq 15$ .

$15 \leq y < 38$

Solve $- y + 15 < 23$ when $y < 15$ .

Solve $- y + 15 < 23$ for $y$ .

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

Subtract $15$ from both sides of the inequality.

$- y < 23 - 15$

Subtract $15$ from $23$ .

$- y < 8$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $8$ by $- 1$ .

$y > - 8$

Find the intersection of $y > - 8$ and $y < 15$ .

$- 8 < y < 15$

Find the union of the solutions.

$- 8 < y < 38$

The result can be shown in multiple forms.

Inequality Form:

$- 8 < y < 38$

Interval Notation:

$(- 8, 38)$

Solve for y |y-15|<23

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