Precalculus Examples

$4 x + 3 y \leq - 62 x - y \leq - 8$

Step 1

Solve $4 x + 3 y \leq - 62 x - y$ for $y$ .

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Step 1.1

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

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Step 1.1.1

Add $y$ to both sides of the inequality.

$4 x + 3 y + y \leq - 62 x$

Step 1.1.2

Add $3 y$ and $y$ .

$4 x + 4 y \leq - 62 x$

Step 1.2

Move all terms not containing $y$ to the right side of the inequality.

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Step 1.2.1

Subtract $4 x$ from both sides of the inequality.

$4 y \leq - 62 x - 4 x$

Step 1.2.2

Subtract $4 x$ from $- 62 x$ .

$4 y \leq - 66 x$

Step 1.3

Divide each term in $4 y \leq - 66 x$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y \leq - 66 x$ by $4$ .

$\frac{4 y}{4} \leq \frac{- 66 x}{4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} \leq \frac{- 66 x}{4}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- 66 x}{4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Cancel the common factor of $- 66$ and $4$ .

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Step 1.3.3.1.1

Factor $2$ out of $- 66 x$ .

$y \leq \frac{2 (- 33 x)}{4}$

Step 1.3.3.1.2

Cancel the common factors.

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Step 1.3.3.1.2.1

Factor $2$ out of $4$ .

$y \leq \frac{2 (- 33 x)}{2 (2)}$

Step 1.3.3.1.2.2

Cancel the common factor.

$y \leq \frac{2 (- 33 x)}{2 \cdot 2}$

Step 1.3.3.1.2.3

Rewrite the expression.

$y \leq \frac{- 33 x}{2}$

Step 1.3.3.2

Move the negative in front of the fraction.

$y \leq - \frac{33 x}{2}$

Step 2

Solve $- 62 x - y \leq - 8$ for $y$ .

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Step 2.1

Add $62 x$ to both sides of the inequality.

$- y \leq - 8 + 62 x$

Step 2.2

Divide each term in $- y \leq - 8 + 62 x$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- y \leq - 8 + 62 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{- 8}{- 1} + \frac{62 x}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{- 8}{- 1} + \frac{62 x}{- 1}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y \geq \frac{- 8}{- 1} + \frac{62 x}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $- 8$ by $- 1$ .

$y \geq 8 + \frac{62 x}{- 1}$

Step 2.2.3.1.2

Move the negative one from the denominator of $\frac{62 x}{- 1}$ .

$y \geq 8 - 1 \cdot (62 x)$

Step 2.2.3.1.3

Rewrite $- 1 \cdot (62 x)$ as $- (62 x)$ .

$y \geq 8 - (62 x)$

Step 2.2.3.1.4

Multiply $62$ by $- 1$ .

$y \geq 8 - 62 x$

Step 3

Find the intersection of $y \leq - \frac{33 x}{2}$ and $y \geq 8 - 62 x$ .

$8 - 62 x \leq y \leq - \frac{33 x}{2}$

Step 4