Solve for y |3-9y|<=33

$| 3 - 9 y | \leq 33$

Step 1

Write $| 3 - 9 y | \leq 33$ as a piecewise.

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

$3 - 9 y \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 9 y \geq - 3$

Divide each term in $- 9 y \geq - 3$ by $- 9$ and simplify.

Divide each term in $- 9 y \geq - 3$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 y}{- 9} \leq \frac{- 3}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 y}{- 9} \leq \frac{- 3}{- 9}$

Divide $y$ by $1$ .

$y \leq \frac{- 3}{- 9}$

Simplify the right side.

Cancel the common factor of $- 3$ and $- 9$ .

Factor $- 3$ out of $- 3$ .

$y \leq \frac{- 3 (1)}{- 9}$

Cancel the common factors.

Factor $- 3$ out of $- 9$ .

$y \leq \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Cancel the common factor.

$y \leq \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Rewrite the expression.

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

In the piece where $3 - 9 y$ is non-negative, remove the absolute value.

$3 - 9 y \leq 33$

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

$3 - 9 y < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 9 y < - 3$

Divide each term in $- 9 y < - 3$ by $- 9$ and simplify.

Divide each term in $- 9 y < - 3$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 y}{- 9} > \frac{- 3}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 y}{- 9} > \frac{- 3}{- 9}$

Divide $y$ by $1$ .

$y > \frac{- 3}{- 9}$

Simplify the right side.

Cancel the common factor of $- 3$ and $- 9$ .

Factor $- 3$ out of $- 3$ .

$y > \frac{- 3 (1)}{- 9}$

Cancel the common factors.

Factor $- 3$ out of $- 9$ .

$y > \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Cancel the common factor.

$y > \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Rewrite the expression.

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

In the piece where $3 - 9 y$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 - 9 y) \leq 33$

Write as a piecewise.

${\begin{cases} 3 - 9 y \leq 33 & y \leq \frac{1}{3} \\ - (3 - 9 y) \leq 33 & y > \frac{1}{3} \end{cases}$

Simplify $- (3 - 9 y) \leq 33$ .

Apply the distributive property.

${\begin{cases} 3 - 9 y \leq 33 & y \leq \frac{1}{3} \\ - 1 \cdot 3 - (- 9 y) \leq 33 & y > \frac{1}{3} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 - 9 y \leq 33 & y \leq \frac{1}{3} \\ - 3 - (- 9 y) \leq 33 & y > \frac{1}{3} \end{cases}$

Multiply $- 9$ by $- 1$ .

${\begin{cases} 3 - 9 y \leq 33 & y \leq \frac{1}{3} \\ - 3 + 9 y \leq 33 & y > \frac{1}{3} \end{cases}$

Step 2

Solve $3 - 9 y \leq 33$ when $y \leq \frac{1}{3}$ .

Solve $3 - 9 y \leq 33$ for $y$ .

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

Subtract $3$ from both sides of the inequality.

$- 9 y \leq 33 - 3$

Subtract $3$ from $33$ .

$- 9 y \leq 30$

Divide each term in $- 9 y \leq 30$ by $- 9$ and simplify.

Divide each term in $- 9 y \leq 30$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 y}{- 9} \geq \frac{30}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 y}{- 9} \geq \frac{30}{- 9}$

Divide $y$ by $1$ .

$y \geq \frac{30}{- 9}$

Simplify the right side.

Cancel the common factor of $30$ and $- 9$ .

Factor $3$ out of $30$ .

$y \geq \frac{3 (10)}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

$y \geq \frac{3 \cdot 10}{3 \cdot - 3}$

Cancel the common factor.

$y \geq \frac{3 \cdot 10}{3 \cdot - 3}$

Rewrite the expression.

$y \geq \frac{10}{- 3}$

Move the negative in front of the fraction.

$y \geq - \frac{10}{3}$

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

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

Step 3

Solve $- 3 + 9 y \leq 33$ when $y > \frac{1}{3}$ .

Solve $- 3 + 9 y \leq 33$ for $y$ .

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

Add $3$ to both sides of the inequality.

$9 y \leq 33 + 3$

Add $33$ and $3$ .

$9 y \leq 36$

Divide each term in $9 y \leq 36$ by $9$ and simplify.

Divide each term in $9 y \leq 36$ by $9$ .

$\frac{9 y}{9} \leq \frac{36}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} \leq \frac{36}{9}$

Divide $y$ by $1$ .

$y \leq \frac{36}{9}$

Simplify the right side.

Divide $36$ by $9$ .

$y \leq 4$

Find the intersection of $y \leq 4$ and $y > \frac{1}{3}$ .

$\frac{1}{3} < y \leq 4$

Step 4

Find the union of the solutions.

$- \frac{10}{3} \leq y \leq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{10}{3} \leq y \leq 4$

Interval Notation:

$[- \frac{10}{3}, 4]$

Step 6

Solve for y |3-9y|<=33

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