Algebra Examples

$y < 3 x + 2$ and $2 y \geq 6 x - 4$

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 $2 y \geq 6 x - 4$

Step 1.2

Subtract $2$ from both sides of the inequality.

$3 x > y - 2$ and $2 y \geq 6 x - 4$

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 $2 y \geq 6 x - 4$

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 $2 y \geq 6 x - 4$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x > \frac{y}{3} + \frac{- 2}{3}$ and $2 y \geq 6 x - 4$

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 $2 y \geq 6 x - 4$

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 $6 x - 4 \leq 2 y$

Step 2.2

Add $4$ to both sides of the inequality.

$x > \frac{y}{3} - \frac{2}{3}$ and $6 x \leq 2 y + 4$

Step 2.3

Divide each term in $6 x \leq 2 y + 4$ by $6$ and simplify.

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Step 2.3.1

Divide each term in $6 x \leq 2 y + 4$ by $6$ .

$x > \frac{y}{3} - \frac{2}{3}$ and $\frac{6 x}{6} \leq \frac{2 y}{6} + \frac{4}{6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$x > \frac{y}{3} - \frac{2}{3}$ and $\frac{6 x}{6} \leq \frac{2 y}{6} + \frac{4}{6}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{2 y}{6} + \frac{4}{6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

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

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Step 2.3.3.1.1.1

Factor $2$ out of $2 y$ .

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{2 (y)}{6} + \frac{4}{6}$

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $2$ out of $6$ .

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{2 y}{2 \cdot 3} + \frac{4}{6}$

Step 2.3.3.1.1.2.2

Cancel the common factor.

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{2 y}{2 \cdot 3} + \frac{4}{6}$

Step 2.3.3.1.1.2.3

Rewrite the expression.

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{y}{3} + \frac{4}{6}$

Step 2.3.3.1.2

Cancel the common factor of $4$ and $6$ .

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Step 2.3.3.1.2.1

Factor $2$ out of $4$ .

$x > \frac{y}{3} - \frac{2}{3}$ and $x \leq \frac{y}{3} + \frac{2 (2)}{6}$

Step 2.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.1.2.2.1

Factor $2$ out of $6$ .

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

Step 2.3.3.1.2.2.2

Cancel the common factor.

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

Step 2.3.3.1.2.2.3

Rewrite the expression.

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

Step 3