Algebra Examples

$- 1 + 9 y > - 73$ and $\frac{y - 4}{- 3} > 2$

Step 1

Simplify the first inequality.

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Step 1.1

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

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Step 1.1.1

Add $1$ to both sides of the inequality.

$9 y > - 73 + 1$ and $\frac{y - 4}{- 3} > 2$

Step 1.1.2

Add $- 73$ and $1$ .

$9 y > - 72$ and $\frac{y - 4}{- 3} > 2$

Step 1.2

Divide each term in $9 y > - 72$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 y > - 72$ by $9$ .

$\frac{9 y}{9} > \frac{- 72}{9}$ and $\frac{y - 4}{- 3} > 2$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} > \frac{- 72}{9}$ and $\frac{y - 4}{- 3} > 2$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{- 72}{9}$ and $\frac{y - 4}{- 3} > 2$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 72$ by $9$ .

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

Step 2

Simplify the second inequality.

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Step 2.1

Multiply both sides by $- 3$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Simplify terms.

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Step 2.2.1.1.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.1.1

Factor $3$ out of $- 3$ .

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

Step 2.2.1.1.1.1.2

Factor $3$ out of $- 3$ .

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

Step 2.2.1.1.1.1.3

Cancel the common factor.

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

Step 2.2.1.1.1.1.4

Rewrite the expression.

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

Step 2.2.1.1.1.2

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

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

Step 2.2.1.1.1.3

Simplify the expression.

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Step 2.2.1.1.1.3.1

Move the negative one from the denominator of $\frac{(y - 4) \cdot - 1}{- 1}$ .

$y > - 8$ and $- 1 \cdot ((y - 4) \cdot - 1) < 2 \cdot - 3$

Step 2.2.1.1.1.3.2

Rewrite $- 1 \cdot ((y - 4) \cdot - 1)$ as $- ((y - 4) \cdot - 1)$ .

$y > - 8$ and $- ((y - 4) \cdot - 1) < 2 \cdot - 3$

Step 2.2.1.1.1.4

Apply the distributive property.

$y > - 8$ and $- (y \cdot - 1 - 4 \cdot - 1) < 2 \cdot - 3$

Step 2.2.1.1.1.5

Simplify the expression.

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Step 2.2.1.1.1.5.1

Move $- 1$ to the left of $y$ .

$y > - 8$ and $- (- 1 \cdot y - 4 \cdot - 1) < 2 \cdot - 3$

Step 2.2.1.1.1.5.2

Multiply $- 4$ by $- 1$ .

$y > - 8$ and $- (- 1 \cdot y + 4) < 2 \cdot - 3$

Step 2.2.1.1.2

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

$y > - 8$ and $- (- y + 4) < 2 \cdot - 3$

Step 2.2.1.1.3

Apply the distributive property.

$y > - 8$ and $y - 1 \cdot 4 < 2 \cdot - 3$

Step 2.2.1.1.4

Multiply $- - y$ .

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Step 2.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$y > - 8$ and $1 y - 1 \cdot 4 < 2 \cdot - 3$

Step 2.2.1.1.4.2

Multiply $y$ by $1$ .

$y > - 8$ and $y - 1 \cdot 4 < 2 \cdot - 3$

Step 2.2.1.1.5

Multiply $- 1$ by $4$ .

$y > - 8$ and $y - 4 < 2 \cdot - 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $2$ by $- 3$ .

$y > - 8$ and $y - 4 < - 6$

Step 2.3

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

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Step 2.3.1

Add $4$ to both sides of the inequality.

$y > - 8$ and $y < - 6 + 4$

Step 2.3.2

Add $- 6$ and $4$ .

$y > - 8$ and $y < - 2$

Step 3

The intersection consists of the elements that are contained in both intervals.

$- 8 < y < - 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 8 < y < - 2$

Interval Notation:

$(- 8, - 2)$

Step 5