Solve for y |y-2|<10

$| y - 2 | < 10$

Write $| y - 2 | < 10$ as a piecewise.

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

$y - 2 \geq 0$

Add $2$ to both sides of the inequality.

$y \geq 2$

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

$y - 2 < 10$

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

$y - 2 < 0$

Add $2$ to both sides of the inequality.

$y < 2$

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

$- (y - 2) < 10$

Write as a piecewise.

${\begin{cases} y - 2 < 10 & y \geq 2 \\ - (y - 2) < 10 & y < 2 \end{cases}$

Simplify $- (y - 2) < 10$ .

Apply the distributive property.

${\begin{cases} y - 2 < 10 & y \geq 2 \\ - y - - 2 < 10 & y < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} y - 2 < 10 & y \geq 2 \\ - y + 2 < 10 & y < 2 \end{cases}$

Solve $y - 2 < 10$ when $y \geq 2$ .

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

Add $2$ to both sides of the inequality.

$y < 10 + 2$

Add $10$ and $2$ .

$y < 12$

Find the intersection of $y < 12$ and $y \geq 2$ .

$2 \leq y < 12$

Solve $- y + 2 < 10$ when $y < 2$ .

Solve $- y + 2 < 10$ for $y$ .

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

Subtract $2$ from both sides of the inequality.

$- y < 10 - 2$

Subtract $2$ from $10$ .

$- y < 8$

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 $y < 2$ .

$- 8 < y < 2$

Find the union of the solutions.

$- 8 < y < 12$

The result can be shown in multiple forms.

Inequality Form:

$- 8 < y < 12$

Interval Notation:

$(- 8, 12)$

Solve for y |y-2|<10

Related Questions

Support LCMGCF.com by disabling your adblocker.