Algebra Examples

$y < 3 x - 1 y > | x - 2 |$

Step 1

Solve $y < 3 x - 1 y$ for $y$ .

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Step 1.1

Rewrite $- 1 y$ as $- y$ .

$y < 3 x - y$

Step 1.2

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

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Step 1.2.1

Add $y$ to both sides of the inequality.

$y + y < 3 x$

Step 1.2.2

Add $y$ and $y$ .

$2 y < 3 x$

Step 1.3

Divide each term in $2 y < 3 x$ by $2$ and simplify.

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Step 1.3.1

Divide each term in $2 y < 3 x$ by $2$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 2

Solve $3 x - 1 y > | x - 2 |$ for $y$ .

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Step 2.1

Rewrite $- 1 y$ as $- y$ .

$3 x - y > | x - 2 |$

Step 2.2

Subtract $3 x$ from both sides of the inequality.

$- y > | x - 2 | - 3 x$

Step 2.3

Divide each term in $- y > | x - 2 | - 3 x$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- y > | x - 2 | - 3 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{| x - 2 |}{- 1} + \frac{- 3 x}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{| x - 2 |}{- 1} + \frac{- 3 x}{- 1}$

Step 2.3.2.2

Divide $y$ by $1$ .

$y < \frac{| x - 2 |}{- 1} + \frac{- 3 x}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

To write $\frac{| x - 2 |}{- 1}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 2.3.3.2

To write $\frac{- 3 x}{- 1}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 2.3.3.3

Write each expression with a common denominator of $1$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.3.1

Multiply $\frac{| x - 2 |}{- 1}$ by $\frac{- 1}{- 1}$ .

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

Step 2.3.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.3.3

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

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

Step 2.3.3.3.4

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.4

Combine the numerators over the common denominator.

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

Step 2.3.3.5

Simplify each term.

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Step 2.3.3.5.1

Move $- 1$ to the left of $| x - 2 |$ .

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

Step 2.3.3.5.2

Rewrite $- 1 | x - 2 |$ as $- | x - 2 |$ .

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

Step 2.3.3.5.3

Multiply $- 1$ by $- 3$ .

$y < \frac{- | x - 2 | + 3 x}{1}$

Step 2.3.3.6

Divide $- | x - 2 | + 3 x$ by $1$ .

$y < - | x - 2 | + 3 x$

Step 3

Find the intersection of $y < \frac{3 x}{2}$ and $y < - | x - 2 | + 3 x$ .

$y < N o (Min i m u m)$

Step 4