Precalculus Examples

$0 < 2 - \frac{1}{3} y < 4$

Step 1

Combine $y$ and $\frac{1}{3}$ .

$0 < 2 - \frac{y}{3} < 4$

Step 2

Move all terms not containing $y$ from the center section of the inequality.

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Step 2.1

Subtract $2$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$0 - 2 < - \frac{y}{3} < 4 - 2$

Step 2.2

Subtract $2$ from $0$ .

$- 2 < - \frac{y}{3} < 4 - 2$

Step 2.3

Multiply each term in the inequality by $3$ .

$- 2 \cdot 3 < - \frac{y}{3} \cdot 3 < 4 \cdot 3 - 2 \cdot 3$

Step 2.4

Multiply $- 2$ by $3$ .

$- 6 < - \frac{y}{3} \cdot 3 < 4 \cdot 3 - 2 \cdot 3$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Move the leading negative in $- \frac{y}{3}$ into the numerator.

$- 6 < \frac{- y}{3} \cdot 3 < 4 \cdot 3 - 2 \cdot 3$

Step 2.5.2

Cancel the common factor.

$- 6 < \frac{- y}{3} \cdot 3 < 4 \cdot 3 - 2 \cdot 3$

Step 2.5.3

Rewrite the expression.

$- 6 < - y < 4 \cdot 3 - 2 \cdot 3$

Step 2.6

Simplify each term.

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Step 2.6.1

Multiply $4$ by $3$ .

$- 6 < - y < 12 - 2 \cdot 3$

Step 2.6.2

Multiply $- 2$ by $3$ .

$- 6 < - y < 12 - 6$

Step 2.7

Subtract $6$ from $12$ .

$- 6 < - y < 6$

Step 3

Multiply each term in the inequality by $- 1$ .

$\frac{- 6}{- 1} > \frac{- y}{- 1} > \frac{6}{- 1}$

Step 4

Divide $- 6$ by $- 1$ .

$6 > \frac{- y}{- 1} > \frac{6}{- 1}$

Step 5

Dividing two negative values results in a positive value.

$6 > \frac{y}{1} > \frac{6}{- 1}$

Step 6

Divide $y$ by $1$ .

$6 > y > \frac{6}{- 1}$

Step 7

Divide $6$ by $- 1$ .

$6 > y > - 6$

Step 8

Rewrite the interval so that the left value is less than the right value. This is the correct way to write an interval solution.

$- 6 < y < 6$

Step 9

Convert the inequality to interval notation.

$(- 6, 6)$

Step 10