Algebra Examples

$y > x - 4 y < - | x - 2 |$

Step 1

Solve $y > x - 4 y$ 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 $4 y$ to both sides of the inequality.

$y + 4 y > x$

Step 1.1.2

Add $y$ and $4 y$ .

$5 y > x$

Step 1.2

Divide each term in $5 y > x$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 y > x$ by $5$ .

$\frac{5 y}{5} > \frac{x}{5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} > \frac{x}{5}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{x}{5}$

Step 2

Solve $x - 4 y < - | x - 2 |$ for $y$ .

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Step 2.1

Subtract $x$ from both sides of the inequality.

$- 4 y < - | x - 2 | - x$

Step 2.2

Divide each term in $- 4 y < - | x - 2 | - x$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 y < - | x - 2 | - x$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 y}{- 4} > \frac{- | x - 2 |}{- 4} + \frac{- x}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} > \frac{- | x - 2 |}{- 4} + \frac{- x}{- 4}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{- | x - 2 |}{- 4} + \frac{- x}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

To write $\frac{- | x - 2 |}{- 4}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$y > \frac{- | x - 2 |}{- 4} \cdot \frac{- 1}{- 1} + \frac{- x}{- 4}$

Step 2.2.3.2

To write $\frac{- x}{- 4}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$y > \frac{- | x - 2 |}{- 4} \cdot \frac{- 1}{- 1} + \frac{- x}{- 4} \cdot \frac{- 1}{- 1}$

Step 2.2.3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.3.3.1

Multiply $\frac{- | x - 2 |}{- 4}$ by $\frac{- 1}{- 1}$ .

$y > \frac{- | x - 2 | \cdot - 1}{- 4 \cdot - 1} + \frac{- x}{- 4} \cdot \frac{- 1}{- 1}$

Step 2.2.3.3.2

Multiply $- 4$ by $- 1$ .

$y > \frac{- | x - 2 | \cdot - 1}{4} + \frac{- x}{- 4} \cdot \frac{- 1}{- 1}$

Step 2.2.3.3.3

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

$y > \frac{- | x - 2 | \cdot - 1}{4} + \frac{- x \cdot - 1}{- 4 \cdot - 1}$

Step 2.2.3.3.4

Multiply $- 4$ by $- 1$ .

$y > \frac{- | x - 2 | \cdot - 1}{4} + \frac{- x \cdot - 1}{4}$

Step 2.2.3.4

Combine the numerators over the common denominator.

$y > \frac{- | x - 2 | \cdot - 1 - x \cdot - 1}{4}$

Step 2.2.3.5

Simplify each term.

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Step 2.2.3.5.1

Multiply $- | x - 2 | \cdot - 1$ .

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Step 2.2.3.5.1.1

Multiply $- 1$ by $- 1$ .

$y > \frac{1 | x - 2 | - x \cdot - 1}{4}$

Step 2.2.3.5.1.2

Multiply $| x - 2 |$ by $1$ .

$y > \frac{| x - 2 | - x \cdot - 1}{4}$

Step 2.2.3.5.2

Multiply $- x \cdot - 1$ .

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Step 2.2.3.5.2.1

Multiply $- 1$ by $- 1$ .

$y > \frac{| x - 2 | + 1 x}{4}$

Step 2.2.3.5.2.2

Multiply $x$ by $1$ .

$y > \frac{| x - 2 | + x}{4}$

Step 3

Find the intersection of $y > \frac{x}{5}$ and $y > \frac{| x - 2 | + x}{4}$ .

$y > \frac{x}{5}$ and $y > \frac{| x - 2 | + x}{4}$

Step 4