Solve the Absolute Value Inequality for y |y-4|>=8

$| y - 4 | \geq 8$

Write $| y - 4 | \geq 8$ as a piecewise.

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

$y - 4 \geq 0$

Add $4$ to both sides of the inequality.

$y \geq 4$

In the piece where $y - 4$ is non-negative, remove the absolute value.

$y - 4 \geq 8$

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

$y - 4 < 0$

Add $4$ to both sides of the inequality.

$y < 4$

In the piece where $y - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y - 4) \geq 8$

Write as a piecewise.

${\begin{cases} y - 4 \geq 8 & y \geq 4 \\ - (y - 4) \geq 8 & y < 4 \end{cases}$

Simplify $- (y - 4) \geq 8$ .

Apply the distributive property.

${\begin{cases} y - 4 \geq 8 & y \geq 4 \\ - y - - 4 \geq 8 & y < 4 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} y - 4 \geq 8 & y \geq 4 \\ - y + 4 \geq 8 & y < 4 \end{cases}$

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

Add $4$ to both sides of the inequality.

$y \geq 8 + 4$

Add $8$ and $4$ .

$y \geq 12$

Solve $- y + 4 \geq 8$ for $y$ .

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

Subtract $4$ from both sides of the inequality.

$- y \geq 8 - 4$

Subtract $4$ from $8$ .

$- y \geq 4$

Divide each term in $- y \geq 4$ by $- 1$ and simplify.

Divide each term in $- y \geq 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{4}{- 1}$

Divide $y$ by $1$ .

$y \leq \frac{4}{- 1}$

Simplify the right side.

Divide $4$ by $- 1$ .

$y \leq - 4$

Find the union of the solutions.

$y \leq - 4$ or $y \geq 12$

The result can be shown in multiple forms.

Inequality Form:

$y \leq - 4 or y \geq 12$

Interval Notation:

$(- \infty, - 4] \cup [12, \infty)$

Solve the Absolute Value Inequality for y |y-4|>=8

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