Algebra Examples

- 1 + 9 y > - 73

$- 1 + 9 y > - 73$ and

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

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

Step 1

Simplify the first inequality.

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Step 1.1

Move all terms not containing

y

$y$ to the right side of the inequality.

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Step 1.1.1

Add

1

$1$ to both sides of the inequality.

9 y > - 73 + 1

$9 y > - 73 + 1$ and

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

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

Step 1.1.2

Add

- 73

$- 73$ and

1

$1$ .

9 y > - 72

$9 y > - 72$ and

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

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

9 y > - 72

$9 y > - 72$ and

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

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

Step 1.2

Divide each term in

9 y > - 72

$9 y > - 72$ by

9

$9$ and simplify.

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Step 1.2.1

Divide each term in

9 y > - 72

$9 y > - 72$ by

9

$9$ .

\frac{9 y}{9} > \frac{- 72}{9}

$\frac{9 y}{9} > \frac{- 72}{9}$ and

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

9

$9$ .

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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

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

$y > \frac{- 72}{9}$ and

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

$y > \frac{- 72}{9}$ 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

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

$y > - 8$ and

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

$y > - 8$ and

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

$y > - 8$ 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

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

$y > - 8$ 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

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

$y > - 8$ 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

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

$y > - 8$ and

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

$y > - 8$ 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

$y - 1 \cdot 4 < 2 \cdot - 3$

$y > - 8$ and

$y - 1 \cdot 4 < 2 \cdot - 3$

Step 2.2.1.1.5

Multiply

$- 1$ by

$4$ .

$y > - 8$ and

$y - 4 < 2 \cdot - 3$

$y > - 8$ and

$y - 4 < 2 \cdot - 3$

$y > - 8$ 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

$y - 4 < - 6$

$y > - 8$ and

$y - 4 < - 6$

$y > - 8$ 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

$y < - 2$

$y > - 8$ and

$y < - 2$

$y > - 8$ 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