Algebra Examples

$\frac{y - 1}{2} - \frac{2 y + 3}{8} - y > 2$

Step 1

Simplify $\frac{y - 1}{2} - \frac{2 y + 3}{8} - y$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

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

$\frac{y - 1}{2} \cdot \frac{4}{4} - \frac{2 y + 3}{8} - y > 2$

Step 1.1.2

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

$\frac{(y - 1) \cdot 4}{2 \cdot 4} - \frac{2 y + 3}{8} - y > 2$

Step 1.1.3

Write $- y$ as a fraction with denominator $1$ .

$\frac{(y - 1) \cdot 4}{2 \cdot 4} - \frac{2 y + 3}{8} + \frac{- y}{1} > 2$

Step 1.1.4

Multiply $\frac{- y}{1}$ by $\frac{8}{8}$ .

$\frac{(y - 1) \cdot 4}{2 \cdot 4} - \frac{2 y + 3}{8} + \frac{- y}{1} \cdot \frac{8}{8} > 2$

Step 1.1.5

Multiply $\frac{- y}{1}$ by $\frac{8}{8}$ .

$\frac{(y - 1) \cdot 4}{2 \cdot 4} - \frac{2 y + 3}{8} + \frac{- y \cdot 8}{8} > 2$

Step 1.1.6

Multiply $2$ by $4$ .

$\frac{(y - 1) \cdot 4}{8} - \frac{2 y + 3}{8} + \frac{- y \cdot 8}{8} > 2$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(y - 1) \cdot 4 - (2 y + 3) - y \cdot 8}{8} > 2$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{y \cdot 4 - 1 \cdot 4 - (2 y + 3) - y \cdot 8}{8} > 2$

Step 1.3.2

Move $4$ to the left of $y$ .

$\frac{4 \cdot y - 1 \cdot 4 - (2 y + 3) - y \cdot 8}{8} > 2$

Step 1.3.3

Multiply $- 1$ by $4$ .

$\frac{4 \cdot y - 4 - (2 y + 3) - y \cdot 8}{8} > 2$

Step 1.3.4

Apply the distributive property.

$\frac{4 y - 4 - (2 y) - 1 \cdot 3 - y \cdot 8}{8} > 2$

Step 1.3.5

Multiply $2$ by $- 1$ .

$\frac{4 y - 4 - 2 y - 1 \cdot 3 - y \cdot 8}{8} > 2$

Step 1.3.6

Multiply $- 1$ by $3$ .

$\frac{4 y - 4 - 2 y - 3 - y \cdot 8}{8} > 2$

Step 1.3.7

Multiply $8$ by $- 1$ .

$\frac{4 y - 4 - 2 y - 3 - 8 y}{8} > 2$

Step 1.4

Simplify terms.

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Step 1.4.1

Subtract $2 y$ from $4 y$ .

$\frac{2 y - 4 - 3 - 8 y}{8} > 2$

Step 1.4.2

Subtract $8 y$ from $2 y$ .

$\frac{- 6 y - 4 - 3}{8} > 2$

Step 1.4.3

Subtract $3$ from $- 4$ .

$\frac{- 6 y - 7}{8} > 2$

Step 1.4.4

Factor $- 1$ out of $- 6 y$ .

$\frac{- (6 y) - 7}{8} > 2$

Step 1.4.5

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- (6 y) - 1 (7)}{8} > 2$

Step 1.4.6

Factor $- 1$ out of $- (6 y) - 1 (7)$ .

$\frac{- (6 y + 7)}{8} > 2$

Step 1.4.7

Simplify the expression.

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Step 1.4.7.1

Rewrite $- (6 y + 7)$ as $- 1 (6 y + 7)$ .

$\frac{- 1 (6 y + 7)}{8} > 2$

Step 1.4.7.2

Move the negative in front of the fraction.

$- \frac{6 y + 7}{8} > 2$

Step 2

Divide each term in $- \frac{6 y + 7}{8} > 2$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{6 y + 7}{8} > 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{6 y + 7}{8}}{- 1} < \frac{2}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{6 y + 7}{8}}{1} < \frac{2}{- 1}$

Step 2.2.2

Divide $\frac{6 y + 7}{8}$ by $1$ .

$\frac{6 y + 7}{8} < \frac{2}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $2$ by $- 1$ .

$\frac{6 y + 7}{8} < - 2$

Step 3

Multiply both sides by $8$ .

$\frac{6 y + 7}{8} \cdot 8 < - 2 \cdot 8$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $8$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{6 y + 7}{8} \cdot 8 < - 2 \cdot 8$

Step 4.1.1.2

Rewrite the expression.

$6 y + 7 < - 2 \cdot 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 2$ by $8$ .

$6 y + 7 < - 16$

Step 5

Solve for $y$ .

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Step 5.1

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

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Step 5.1.1

Subtract $7$ from both sides of the inequality.

$6 y < - 16 - 7$

Step 5.1.2

Subtract $7$ from $- 16$ .

$6 y < - 23$

Step 5.2

Divide each term in $6 y < - 23$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 y < - 23$ by $6$ .

$\frac{6 y}{6} < \frac{- 23}{6}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} < \frac{- 23}{6}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{- 23}{6}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$y < - \frac{23}{6}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$y < - \frac{23}{6}$

Interval Notation:

$(- \infty, - \frac{23}{6})$