Calculus Examples

$4 (2 y - 3) < 2 (3 y + 1) - (5 - 3 y)$

Step 1

Rewrite so $y$ is on the left side of the inequality.

$2 (3 y + 1) - (5 - 3 y) > 4 (2 y - 3)$

Step 2

Simplify $2 (3 y + 1) - (5 - 3 y)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$2 (3 y) + 2 \cdot 1 - (5 - 3 y) > 4 (2 y - 3)$

Step 2.1.2

Multiply $3$ by $2$ .

$6 y + 2 \cdot 1 - (5 - 3 y) > 4 (2 y - 3)$

Step 2.1.3

Multiply $2$ by $1$ .

$6 y + 2 - (5 - 3 y) > 4 (2 y - 3)$

Step 2.1.4

Apply the distributive property.

$6 y + 2 - 1 \cdot 5 - (- 3 y) > 4 (2 y - 3)$

Step 2.1.5

Multiply $- 1$ by $5$ .

$6 y + 2 - 5 - (- 3 y) > 4 (2 y - 3)$

Step 2.1.6

Multiply $- 3$ by $- 1$ .

$6 y + 2 - 5 + 3 y > 4 (2 y - 3)$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $6 y$ and $3 y$ .

$9 y + 2 - 5 > 4 (2 y - 3)$

Step 2.2.2

Subtract $5$ from $2$ .

$9 y - 3 > 4 (2 y - 3)$

Step 3

Simplify $4 (2 y - 3)$ .

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Step 3.1

Apply the distributive property.

$9 y - 3 > 4 (2 y) + 4 \cdot - 3$

Step 3.2

Multiply.

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Step 3.2.1

Multiply $2$ by $4$ .

$9 y - 3 > 8 y + 4 \cdot - 3$

Step 3.2.2

Multiply $4$ by $- 3$ .

$9 y - 3 > 8 y - 12$

Step 4

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

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Step 4.1

Subtract $8 y$ from both sides of the inequality.

$9 y - 3 - 8 y > - 12$

Step 4.2

Subtract $8 y$ from $9 y$ .

$y - 3 > - 12$

Step 5

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

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Step 5.1

Add $3$ to both sides of the inequality.

$y > - 12 + 3$

Step 5.2

Add $- 12$ and $3$ .

$y > - 9$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$y > - 9$

Interval Notation:

$(- 9, \infty)$