Solve the System of @WORD y<1/2x-2y<=-2x+4

$y < \frac{1}{2} x - 2 y \leq - 2 x + 4$

Step 1

Solve $y < \frac{1}{2} x - 2 y$ for $y$ .

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

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

Move all terms containing $y$ to the left side of the inequality.

Add $2 y$ to both sides of the inequality.

$y + 2 y < \frac{x}{2}$

Add $y$ and $2 y$ .

$3 y < \frac{x}{2}$

Divide each term in $3 y < \frac{x}{2}$ by $3$ and simplify.

Divide each term in $3 y < \frac{x}{2}$ by $3$ .

$\frac{3 y}{3} < \frac{\frac{x}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} < \frac{\frac{x}{2}}{3}$

Divide $y$ by $1$ .

$y < \frac{\frac{x}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$y < \frac{x}{2} \cdot \frac{1}{3}$

Multiply $\frac{x}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{x}{2}$ by $\frac{1}{3}$ .

$y < \frac{x}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y < \frac{x}{6}$

Step 2

Solve $\frac{1}{2} x - 2 y \leq - 2 x + 4$ for $y$ .

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

$\frac{x}{2} - 2 y \leq - 2 x + 4$

Subtract $\frac{x}{2}$ from both sides of the inequality.

$- 2 y \leq - 2 x + 4 - \frac{x}{2}$

Divide each term in $- 2 y \leq - 2 x + 4 - \frac{x}{2}$ by $- 2$ and simplify.

Divide each term in $- 2 y \leq - 2 x + 4 - \frac{x}{2}$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} \geq \frac{- 2 x}{- 2} + \frac{4}{- 2} + \frac{- \frac{x}{2}}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} \geq \frac{- 2 x}{- 2} + \frac{4}{- 2} + \frac{- \frac{x}{2}}{- 2}$

Divide $y$ by $1$ .

$y \geq \frac{- 2 x}{- 2} + \frac{4}{- 2} + \frac{- \frac{x}{2}}{- 2}$

Simplify the right side.

Combine the numerators over the common denominator.

$y \geq \frac{- 2 x + 4 - \frac{x}{2}}{- 2}$

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y \geq \frac{- 2 x \cdot \frac{2}{2} - \frac{x}{2} + 4}{- 2}$

Simplify terms.

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

$y \geq \frac{\frac{- 2 x \cdot 2}{2} - \frac{x}{2} + 4}{- 2}$

Combine the numerators over the common denominator.

$y \geq \frac{\frac{- 2 x \cdot 2 - x}{2} + 4}{- 2}$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $- 2 x \cdot 2 - x$ .

Factor $x$ out of $- 2 x \cdot 2$ .

$y \geq \frac{\frac{x (- 2 \cdot 2) - x}{2} + 4}{- 2}$

Factor $x$ out of $- x$ .

$y \geq \frac{\frac{x (- 2 \cdot 2) + x \cdot - 1}{2} + 4}{- 2}$

Factor $x$ out of $x (- 2 \cdot 2) + x \cdot - 1$ .

$y \geq \frac{\frac{x (- 2 \cdot 2 - 1)}{2} + 4}{- 2}$

Multiply $- 2$ by $2$ .

$y \geq \frac{\frac{x (- 4 - 1)}{2} + 4}{- 2}$

Subtract $1$ from $- 4$ .

$y \geq \frac{\frac{x \cdot - 5}{2} + 4}{- 2}$

Move $- 5$ to the left of $x$ .

$y \geq \frac{\frac{- 5 \cdot x}{2} + 4}{- 2}$

Move the negative in front of the fraction.

$y \geq \frac{- \frac{5 x}{2} + 4}{- 2}$

Simplify the numerator.

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y \geq \frac{- \frac{5 x}{2} + 4 \cdot \frac{2}{2}}{- 2}$

Combine $4$ and $\frac{2}{2}$ .

$y \geq \frac{- \frac{5 x}{2} + \frac{4 \cdot 2}{2}}{- 2}$

Combine the numerators over the common denominator.

$y \geq \frac{\frac{- 5 x + 4 \cdot 2}{2}}{- 2}$

Multiply $4$ by $2$ .

$y \geq \frac{\frac{- 5 x + 8}{2}}{- 2}$

Multiply the numerator by the reciprocal of the denominator.

$y \geq \frac{- 5 x + 8}{2} \cdot \frac{1}{- 2}$

Move the negative in front of the fraction.

$y \geq \frac{- 5 x + 8}{2} (- \frac{1}{2})$

Multiply $\frac{- 5 x + 8}{2} (- \frac{1}{2})$ .

Multiply $\frac{- 5 x + 8}{2}$ by $\frac{1}{2}$ .

$y \geq - \frac{- 5 x + 8}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y \geq - \frac{- 5 x + 8}{4}$

Factor $- 1$ out of $- 5 x$ .

$y \geq - \frac{- (5 x) + 8}{4}$

Rewrite $8$ as $- 1 (- 8)$ .

$y \geq - \frac{- (5 x) - 1 (- 8)}{4}$

Factor $- 1$ out of $- (5 x) - 1 (- 8)$ .

$y \geq - \frac{- (5 x - 8)}{4}$

Simplify the expression.

Rewrite $- (5 x - 8)$ as $- 1 (5 x - 8)$ .

$y \geq - \frac{- 1 (5 x - 8)}{4}$

Move the negative in front of the fraction.

$y \geq - - \frac{5 x - 8}{4}$

Multiply $- 1$ by $- 1$ .

$y \geq 1 \frac{5 x - 8}{4}$

Multiply $\frac{5 x - 8}{4}$ by $1$ .

$y \geq \frac{5 x - 8}{4}$

Step 3

Find the intersection of $y < \frac{x}{6}$ and $y \geq \frac{5 x - 8}{4}$ .

$\frac{5 x - 8}{4} \leq y < \frac{x}{6}$

Step 4

Solve the System of @WORD y<1/2x-2y<=-2x+4

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