Solve for y |6y-2|+4<22

$| 6 y - 2 | + 4 < 22$

Step 1

Write $| 6 y - 2 | + 4 < 22$ as a piecewise.

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

$6 y - 2 \geq 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$6 y \geq 2$

Divide each term in $6 y \geq 2$ by $6$ and simplify.

Divide each term in $6 y \geq 2$ by $6$ .

$\frac{6 y}{6} \geq \frac{2}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} \geq \frac{2}{6}$

Divide $y$ by $1$ .

$y \geq \frac{2}{6}$

Simplify the right side.

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$y \geq \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y \geq \frac{2 \cdot 1}{2 \cdot 3}$

Cancel the common factor.

$y \geq \frac{2 \cdot 1}{2 \cdot 3}$

Rewrite the expression.

$y \geq \frac{1}{3}$

In the piece where $6 y - 2$ is non-negative, remove the absolute value.

$6 y - 2 + 4 < 22$

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

$6 y - 2 < 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$6 y < 2$

Divide each term in $6 y < 2$ by $6$ and simplify.

Divide each term in $6 y < 2$ by $6$ .

$\frac{6 y}{6} < \frac{2}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} < \frac{2}{6}$

Divide $y$ by $1$ .

$y < \frac{2}{6}$

Simplify the right side.

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$y < \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y < \frac{2 \cdot 1}{2 \cdot 3}$

Cancel the common factor.

$y < \frac{2 \cdot 1}{2 \cdot 3}$

Rewrite the expression.

$y < \frac{1}{3}$

In the piece where $6 y - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (6 y - 2) + 4 < 22$

Write as a piecewise.

${\begin{cases} 6 y - 2 + 4 < 22 & y \geq \frac{1}{3} \\ - (6 y - 2) + 4 < 22 & y < \frac{1}{3} \end{cases}$

Add $- 2$ and $4$ .

${\begin{cases} 6 y + 2 < 22 & y \geq \frac{1}{3} \\ - (6 y - 2) + 4 < 22 & y < \frac{1}{3} \end{cases}$

Simplify $- (6 y - 2) + 4 < 22$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 6 y + 2 < 22 & y \geq \frac{1}{3} \\ - (6 y) - - 2 + 4 < 22 & y < \frac{1}{3} \end{cases}$

Multiply $6$ by $- 1$ .

${\begin{cases} 6 y + 2 < 22 & y \geq \frac{1}{3} \\ - 6 y - - 2 + 4 < 22 & y < \frac{1}{3} \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 6 y + 2 < 22 & y \geq \frac{1}{3} \\ - 6 y + 2 + 4 < 22 & y < \frac{1}{3} \end{cases}$

Add $2$ and $4$ .

${\begin{cases} 6 y + 2 < 22 & y \geq \frac{1}{3} \\ - 6 y + 6 < 22 & y < \frac{1}{3} \end{cases}$

Step 2

Solve $6 y + 2 < 22$ when $y \geq \frac{1}{3}$ .

Solve $6 y + 2 < 22$ for $y$ .

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

Subtract $2$ from both sides of the inequality.

$6 y < 22 - 2$

Subtract $2$ from $22$ .

$6 y < 20$

Divide each term in $6 y < 20$ by $6$ and simplify.

Divide each term in $6 y < 20$ by $6$ .

$\frac{6 y}{6} < \frac{20}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} < \frac{20}{6}$

Divide $y$ by $1$ .

$y < \frac{20}{6}$

Simplify the right side.

Cancel the common factor of $20$ and $6$ .

Factor $2$ out of $20$ .

$y < \frac{2 (10)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y < \frac{2 \cdot 10}{2 \cdot 3}$

Cancel the common factor.

$y < \frac{2 \cdot 10}{2 \cdot 3}$

Rewrite the expression.

$y < \frac{10}{3}$

Find the intersection of $y < \frac{10}{3}$ and $y \geq \frac{1}{3}$ .

$\frac{1}{3} \leq y < \frac{10}{3}$

Step 3

Solve $- 6 y + 6 < 22$ when $y < \frac{1}{3}$ .

Solve $- 6 y + 6 < 22$ for $y$ .

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

Subtract $6$ from both sides of the inequality.

$- 6 y < 22 - 6$

Subtract $6$ from $22$ .

$- 6 y < 16$

Divide each term in $- 6 y < 16$ by $- 6$ and simplify.

Divide each term in $- 6 y < 16$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 y}{- 6} > \frac{16}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} > \frac{16}{- 6}$

Divide $y$ by $1$ .

$y > \frac{16}{- 6}$

Simplify the right side.

Cancel the common factor of $16$ and $- 6$ .

Factor $2$ out of $16$ .

$y > \frac{2 (8)}{- 6}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$y > \frac{2 \cdot 8}{2 \cdot - 3}$

Cancel the common factor.

$y > \frac{2 \cdot 8}{2 \cdot - 3}$

Rewrite the expression.

$y > \frac{8}{- 3}$

Move the negative in front of the fraction.

$y > - \frac{8}{3}$

Find the intersection of $y > - \frac{8}{3}$ and $y < \frac{1}{3}$ .

$- \frac{8}{3} < y < \frac{1}{3}$

Step 4

Find the union of the solutions.

$- \frac{8}{3} < y < \frac{10}{3}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{8}{3} < y < \frac{10}{3}$

Interval Notation:

$(- \frac{8}{3}, \frac{10}{3})$

Step 6

Solve for y |6y-2|+4<22

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