Algebra Examples

$y \leq - 3 x + 1$ $y \geq - \frac{1}{4} x + 6$

Step 1

Simplify the first inequality.

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Step 1.1

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

$- 3 x + 1 \geq y$

Step 1.2

Subtract $1$ from both sides of the inequality.

$- 3 x \geq y - 1$

Step 1.3

Divide each term in $- 3 x \geq y - 1$ by $- 3$ and simplify.

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Step 1.3.1

Divide each term in $- 3 x \geq y - 1$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{y}{- 3} + \frac{- 1}{- 3}$

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.

$\frac{- 3 x}{- 3} \leq \frac{y}{- 3} + \frac{- 1}{- 3}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{y}{- 3} + \frac{- 1}{- 3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

$x \leq - \frac{y}{3} + \frac{- 1}{- 3}$

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

$x \leq - \frac{y}{3} + \frac{1}{3}$

Step 2

Simplify the second inequality.

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Step 2.1

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

$- \frac{1}{4} x + 6 \leq y$

Step 2.2

Combine $x$ and $\frac{1}{4}$ .

$- \frac{x}{4} + 6 \leq y$

Step 2.3

Subtract $6$ from both sides of the inequality.

$- \frac{x}{4} \leq y - 6$

Step 2.4

Divide each term in $- \frac{x}{4} \leq y - 6$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- \frac{x}{4} \leq y - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{4}}{- 1} \geq \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{4}}{1} \geq \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.4.2.2

Divide $\frac{x}{4}$ by $1$ .

$\frac{x}{4} \geq \frac{y}{- 1} + \frac{- 6}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$\frac{x}{4} \geq - 1 \cdot y + \frac{- 6}{- 1}$

Step 2.4.3.1.2

Rewrite $- 1 \cdot y$ as $- y$ .

$\frac{x}{4} \geq - y + \frac{- 6}{- 1}$

Step 2.4.3.1.3

Divide $- 6$ by $- 1$ .

$\frac{x}{4} \geq - y + 6$

Step 2.5

Multiply both sides by $4$ .

$\frac{x}{4} \cdot 4 \geq (- y + 6) \cdot 4$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $4$ .

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Step 2.6.1.1.1

Cancel the common factor.

$\frac{x}{4} \cdot 4 \geq (- y + 6) \cdot 4$

Step 2.6.1.1.2

Rewrite the expression.

$x \geq (- y + 6) \cdot 4$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $(- y + 6) \cdot 4$ .

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Step 2.6.2.1.1

Apply the distributive property.

$x \geq - y \cdot 4 + 6 \cdot 4$

Step 2.6.2.1.2

Multiply.

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Step 2.6.2.1.2.1

Multiply $4$ by $- 1$ .

$x \geq - 4 y + 6 \cdot 4$

Step 2.6.2.1.2.2

Multiply $6$ by $4$ .

$x \geq - 4 y + 24$

Step 3