Pre-Algebra Examples

$1 - \frac{7}{6} y > \frac{9}{4} - \frac{5}{6} y$

Step 1

Simplify each term.

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Step 1.1

Combine $y$ and $\frac{7}{6}$ .

$1 - \frac{y \cdot 7}{6} > \frac{9}{4} - \frac{5}{6} y$

Step 1.2

Move $7$ to the left of $y$ .

$1 - \frac{7 y}{6} > \frac{9}{4} - \frac{5}{6} y$

Step 2

Simplify each term.

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Step 2.1

Combine $y$ and $\frac{5}{6}$ .

$1 - \frac{7 y}{6} > \frac{9}{4} - \frac{y \cdot 5}{6}$

Step 2.2

Move $5$ to the left of $y$ .

$1 - \frac{7 y}{6} > \frac{9}{4} - \frac{5 y}{6}$

Step 3

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

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Step 3.1

Add $\frac{5 y}{6}$ to both sides of the inequality.

$1 - \frac{7 y}{6} + \frac{5 y}{6} > \frac{9}{4}$

Step 3.2

Combine the numerators over the common denominator.

$1 + \frac{- 7 y + 5 y}{6} > \frac{9}{4}$

Step 3.3

Add $- 7 y$ and $5 y$ .

$1 + \frac{- 2 y}{6} > \frac{9}{4}$

Step 3.4

Simplify each term.

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Step 3.4.1

Cancel the common factor of $- 2$ and $6$ .

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Step 3.4.1.1

Factor $2$ out of $- 2 y$ .

$1 + \frac{2 (- y)}{6} > \frac{9}{4}$

Step 3.4.1.2

Cancel the common factors.

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Step 3.4.1.2.1

Factor $2$ out of $6$ .

$1 + \frac{2 (- y)}{2 (3)} > \frac{9}{4}$

Step 3.4.1.2.2

Cancel the common factor.

$1 + \frac{2 (- y)}{2 \cdot 3} > \frac{9}{4}$

Step 3.4.1.2.3

Rewrite the expression.

$1 + \frac{- y}{3} > \frac{9}{4}$

Step 3.4.2

Move the negative in front of the fraction.

$1 - \frac{y}{3} > \frac{9}{4}$

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the inequality.

$- \frac{y}{3} > \frac{9}{4} - 1$

Step 4.2

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

$- \frac{y}{3} > \frac{9}{4} - 1 \cdot \frac{4}{4}$

Step 4.3

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

$- \frac{y}{3} > \frac{9}{4} + \frac{- 1 \cdot 4}{4}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{y}{3} > \frac{9 - 1 \cdot 4}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $4$ .

$- \frac{y}{3} > \frac{9 - 4}{4}$

Step 4.5.2

Subtract $4$ from $9$ .

$- \frac{y}{3} > \frac{5}{4}$

Step 5

Divide each term in $- \frac{y}{3} > \frac{5}{4}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{y}{3} > \frac{5}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{y}{3}}{- 1} < \frac{\frac{5}{4}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{y}{3}}{1} < \frac{\frac{5}{4}}{- 1}$

Step 5.2.2

Divide $\frac{y}{3}$ by $1$ .

$\frac{y}{3} < \frac{\frac{5}{4}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative one from the denominator of $\frac{\frac{5}{4}}{- 1}$ .

$\frac{y}{3} < - 1 \cdot \frac{5}{4}$

Step 5.3.2

Rewrite $- 1 \cdot \frac{5}{4}$ as $- \frac{5}{4}$ .

$\frac{y}{3} < - \frac{5}{4}$

Step 6

Multiply both sides by $3$ .

$\frac{y}{3} \cdot 3 < - \frac{5}{4} \cdot 3$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{y}{3} \cdot 3 < - \frac{5}{4} \cdot 3$

Step 7.1.1.2

Rewrite the expression.

$y < - \frac{5}{4} \cdot 3$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{5}{4} \cdot 3$ .

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Step 7.2.1.1

Multiply $- \frac{5}{4} \cdot 3$ .

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Step 7.2.1.1.1

Multiply $3$ by $- 1$ .

$y < - 3 (\frac{5}{4})$

Step 7.2.1.1.2

Combine $- 3$ and $\frac{5}{4}$ .

$y < \frac{- 3 \cdot 5}{4}$

Step 7.2.1.1.3

Multiply $- 3$ by $5$ .

$y < \frac{- 15}{4}$

Step 7.2.1.2

Move the negative in front of the fraction.

$y < - \frac{15}{4}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$y < - \frac{15}{4}$

Interval Notation:

$(- \infty, - \frac{15}{4})$