Solve for y |2y+3|<19

$| 2 y + 3 | < 19$

Write $| 2 y + 3 | < 19$ as a piecewise.

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

$2 y + 3 \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$2 y \geq - 3$

Divide each term in $2 y \geq - 3$ by $2$ and simplify.

Divide each term in $2 y \geq - 3$ by $2$ .

$\frac{2 y}{2} \geq \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} \geq \frac{- 3}{2}$

Divide $y$ by $1$ .

$y \geq \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$y \geq - \frac{3}{2}$

In the piece where $2 y + 3$ is non-negative, remove the absolute value.

$2 y + 3 < 19$

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

$2 y + 3 < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$2 y < - 3$

Divide each term in $2 y < - 3$ by $2$ and simplify.

Divide each term in $2 y < - 3$ by $2$ .

$\frac{2 y}{2} < \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} < \frac{- 3}{2}$

Divide $y$ by $1$ .

$y < \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$y < - \frac{3}{2}$

In the piece where $2 y + 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 y + 3) < 19$

Write as a piecewise.

${\begin{cases} 2 y + 3 < 19 & y \geq - \frac{3}{2} \\ - (2 y + 3) < 19 & y < - \frac{3}{2} \end{cases}$

Simplify $- (2 y + 3) < 19$ .

Apply the distributive property.

${\begin{cases} 2 y + 3 < 19 & y \geq - \frac{3}{2} \\ - (2 y) - 1 \cdot 3 < 19 & y < - \frac{3}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 y + 3 < 19 & y \geq - \frac{3}{2} \\ - 2 y - 1 \cdot 3 < 19 & y < - \frac{3}{2} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 2 y + 3 < 19 & y \geq - \frac{3}{2} \\ - 2 y - 3 < 19 & y < - \frac{3}{2} \end{cases}$

Solve $2 y + 3 < 19$ when $y \geq - \frac{3}{2}$ .

Solve $2 y + 3 < 19$ for $y$ .

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

Subtract $3$ from both sides of the inequality.

$2 y < 19 - 3$

Subtract $3$ from $19$ .

$2 y < 16$

Divide each term in $2 y < 16$ by $2$ and simplify.

Divide each term in $2 y < 16$ by $2$ .

$\frac{2 y}{2} < \frac{16}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} < \frac{16}{2}$

Divide $y$ by $1$ .

$y < \frac{16}{2}$

Simplify the right side.

Divide $16$ by $2$ .

$y < 8$

Find the intersection of $y < 8$ and $y \geq - \frac{3}{2}$ .

$- \frac{3}{2} \leq y < 8$

Solve $- 2 y - 3 < 19$ when $y < - \frac{3}{2}$ .

Solve $- 2 y - 3 < 19$ for $y$ .

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

Add $3$ to both sides of the inequality.

$- 2 y < 19 + 3$

Add $19$ and $3$ .

$- 2 y < 22$

Divide each term in $- 2 y < 22$ by $- 2$ and simplify.

Divide each term in $- 2 y < 22$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} > \frac{22}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} > \frac{22}{- 2}$

Divide $y$ by $1$ .

$y > \frac{22}{- 2}$

Simplify the right side.

Divide $22$ by $- 2$ .

$y > - 11$

Find the intersection of $y > - 11$ and $y < - \frac{3}{2}$ .

$- 11 < y < - \frac{3}{2}$

Find the union of the solutions.

$- 11 < y < 8$

The result can be shown in multiple forms.

Inequality Form:

$- 11 < y < 8$

Interval Notation:

$(- 11, 8)$

Solve for y |2y+3|<19

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