Algebra Examples

$y > \frac{1}{2} x - 1$ $y \leq - 2 x + 6$

Step 1

Simplify the first inequality.

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Step 1.1

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

$\frac{1}{2} x - 1 < y$ and $y \leq - 2 x + 6$

Step 1.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} - 1 < y$ and $y \leq - 2 x + 6$

Step 1.3

Add $1$ to both sides of the inequality.

$\frac{x}{2} < y + 1$ and $y \leq - 2 x + 6$

Step 1.4

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 < (y + 1) \cdot 2$ and $y \leq - 2 x + 6$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Cancel the common factor of $2$ .

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Step 1.5.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 < (y + 1) \cdot 2$ and $y \leq - 2 x + 6$

Step 1.5.1.1.2

Rewrite the expression.

$x < (y + 1) \cdot 2$ and $y \leq - 2 x + 6$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify $(y + 1) \cdot 2$ .

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Step 1.5.2.1.1

Apply the distributive property.

$x < y \cdot 2 + 1 \cdot 2$ and $y \leq - 2 x + 6$

Step 1.5.2.1.2

Simplify the expression.

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Step 1.5.2.1.2.1

Move $2$ to the left of $y$ .

$x < 2 \cdot y + 1 \cdot 2$ and $y \leq - 2 x + 6$

Step 1.5.2.1.2.2

Multiply $2$ by $1$ .

$x < 2 y + 2$ and $y \leq - 2 x + 6$

Step 2

Simplify the second inequality.

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Step 2.1

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

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

Step 2.2

Subtract $6$ from both sides of the inequality.

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

Step 2.3

Divide each term in $- 2 x \geq y - 6$ by $- 2$ and simplify.

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Step 2.3.1

Divide each term in $- 2 x \geq y - 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$x < 2 y + 2$ and $\frac{- 2 x}{- 2} \leq \frac{y}{- 2} + \frac{- 6}{- 2}$

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.

$x < 2 y + 2$ and $\frac{- 2 x}{- 2} \leq \frac{y}{- 2} + \frac{- 6}{- 2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x < 2 y + 2$ and $x \leq \frac{y}{- 2} + \frac{- 6}{- 2}$

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

Move the negative in front of the fraction.

$x < 2 y + 2$ and $x \leq - \frac{y}{2} + \frac{- 6}{- 2}$

Step 2.3.3.1.2

Divide $- 6$ by $- 2$ .

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

Step 3