Solve the Absolute Value Inequality for y |y-20|<4

$| y - 20 | < 4$

Step 1

Write $| y - 20 | < 4$ as a piecewise.

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

$y - 20 \geq 0$

Add $20$ to both sides of the inequality.

$y \geq 20$

In the piece where $y - 20$ is non-negative, remove the absolute value.

$y - 20 < 4$

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

$y - 20 < 0$

Add $20$ to both sides of the inequality.

$y < 20$

In the piece where $y - 20$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y - 20) < 4$

Write as a piecewise.

${\begin{cases} y - 20 < 4 & y \geq 20 \\ - (y - 20) < 4 & y < 20 \end{cases}$

Simplify $- (y - 20) < 4$ .

Apply the distributive property.

${\begin{cases} y - 20 < 4 & y \geq 20 \\ - y - - 20 < 4 & y < 20 \end{cases}$

Multiply $- 1$ by $- 20$ .

${\begin{cases} y - 20 < 4 & y \geq 20 \\ - y + 20 < 4 & y < 20 \end{cases}$

Step 2

Solve $y - 20 < 4$ when $y \geq 20$ .

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

Add $20$ to both sides of the inequality.

$y < 4 + 20$

Add $4$ and $20$ .

$y < 24$

Find the intersection of $y < 24$ and $y \geq 20$ .

$20 \leq y < 24$

Step 3

Solve $- y + 20 < 4$ when $y < 20$ .

Solve $- y + 20 < 4$ for $y$ .

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

Subtract $20$ from both sides of the inequality.

$- y < 4 - 20$

Subtract $20$ from $4$ .

$- y < - 16$

Divide each term in $- y < - 16$ by $- 1$ and simplify.

Divide each term in $- y < - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{- 16}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{- 16}{- 1}$

Divide $y$ by $1$ .

$y > \frac{- 16}{- 1}$

Simplify the right side.

Divide $- 16$ by $- 1$ .

$y > 16$

Find the intersection of $y > 16$ and $y < 20$ .

$16 < y < 20$

Step 4

Find the union of the solutions.

$16 < y < 24$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$16 < y < 24$

Interval Notation:

$(16, 24)$

Step 6

Solve the Absolute Value Inequality for y |y-20|<4

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