Algebra Examples

$y > 3 x + 2$ and $y < 2 x + 3$

Step 1

Simplify the first inequality.

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Step 1.1

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

$3 x + 2 < y$ and $y < 2 x + 3$

Step 1.2

Subtract $2$ from both sides of the inequality.

$3 x < y - 2$ and $y < 2 x + 3$

Step 1.3

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

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Step 1.3.1

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

$\frac{3 x}{3} < \frac{y}{3} + \frac{- 2}{3}$ and $y < 2 x + 3$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{y}{3} + \frac{- 2}{3}$ and $y < 2 x + 3$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{y}{3} + \frac{- 2}{3}$ and $y < 2 x + 3$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$x < \frac{y}{3} - \frac{2}{3}$ and $y < 2 x + 3$

Step 2

Simplify the second inequality.

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Step 2.1

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

$x < \frac{y}{3} - \frac{2}{3}$ and $2 x + 3 > y$

Step 2.2

Subtract $3$ from both sides of the inequality.

$x < \frac{y}{3} - \frac{2}{3}$ and $2 x > y - 3$

Step 2.3

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

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Step 2.3.1

Divide each term in $2 x > y - 3$ by $2$ .

$x < \frac{y}{3} - \frac{2}{3}$ and $\frac{2 x}{2} > \frac{y}{2} + \frac{- 3}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$x < \frac{y}{3} - \frac{2}{3}$ and $\frac{2 x}{2} > \frac{y}{2} + \frac{- 3}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{y}{3} - \frac{2}{3}$ and $x > \frac{y}{2} + \frac{- 3}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$x < \frac{y}{3} - \frac{2}{3}$ and $x > \frac{y}{2} - \frac{3}{2}$

Step 3