Solve the System of Inequalities 2<6-2y<5

$2 < 6 - 2 y < 5$

Step 1

Solve $2 < 6 - 2 y$ for $y$ .

Rewrite so $y$ is on the left side of the inequality.

$6 - 2 y > 2$

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

Subtract $6$ from both sides of the inequality.

$- 2 y > 2 - 6$

Subtract $6$ from $2$ .

$- 2 y > - 4$

Divide each term in $- 2 y > - 4$ by $- 2$ and simplify.

Divide each term in $- 2 y > - 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $- 2$ .

$y < 2$

Step 2

Solve $6 - 2 y < 5$ for $y$ .

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

Subtract $6$ from both sides of the inequality.

$- 2 y < 5 - 6$

Subtract $6$ from $5$ .

$- 2 y < - 1$

Divide each term in $- 2 y < - 1$ by $- 2$ and simplify.

Divide each term in $- 2 y < - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} > \frac{- 1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} > \frac{- 1}{- 2}$

Divide $y$ by $1$ .

$y > \frac{- 1}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$y > \frac{1}{2}$

Step 3

Find the intersection of $y < 2$ and $y > \frac{1}{2}$ .

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

Step 4

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(\frac{1}{2}, 2)$

Step 5

Solve the System of Inequalities 2<6-2y<5

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