Precalculus Examples

$\frac{2}{3} y - \frac{1}{2} \cdot (7 - y) < \frac{5 y}{3} - (4 + y)$

Step 1

Simplify $\frac{2}{3} y - \frac{1}{2} \cdot (7 - y)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{2}{3}$ and $y$ .

$\frac{2 y}{3} - \frac{1}{2} \cdot (7 - y) < \frac{5 y}{3} - (4 + y)$

Step 1.1.2

Apply the distributive property.

$\frac{2 y}{3} - \frac{1}{2} \cdot 7 - \frac{1}{2} (- y) < \frac{5 y}{3} - (4 + y)$

Step 1.1.3

Multiply $- \frac{1}{2} \cdot 7$ .

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Step 1.1.3.1

Multiply $7$ by $- 1$ .

$\frac{2 y}{3} - 7 (\frac{1}{2}) - \frac{1}{2} (- y) < \frac{5 y}{3} - (4 + y)$

Step 1.1.3.2

Combine $- 7$ and $\frac{1}{2}$ .

$\frac{2 y}{3} + \frac{- 7}{2} - \frac{1}{2} (- y) < \frac{5 y}{3} - (4 + y)$

Step 1.1.4

Multiply $- \frac{1}{2} (- y)$ .

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Step 1.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 y}{3} + \frac{- 7}{2} + 1 (\frac{1}{2}) y < \frac{5 y}{3} - (4 + y)$

Step 1.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{2 y}{3} + \frac{- 7}{2} + \frac{1}{2} y < \frac{5 y}{3} - (4 + y)$

Step 1.1.4.3

Combine $\frac{1}{2}$ and $y$ .

$\frac{2 y}{3} + \frac{- 7}{2} + \frac{y}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.1.5

Move the negative in front of the fraction.

$\frac{2 y}{3} - \frac{7}{2} + \frac{y}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.2

To write $\frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 y}{3} \cdot \frac{2}{2} + \frac{y}{2} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.3

To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 y}{3} \cdot \frac{2}{2} + \frac{y}{2} \cdot \frac{3}{3} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 y}{3}$ by $\frac{2}{2}$ .

$\frac{2 y \cdot 2}{3 \cdot 2} + \frac{y}{2} \cdot \frac{3}{3} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{2 y \cdot 2}{6} + \frac{y}{2} \cdot \frac{3}{3} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.4.3

Multiply $\frac{y}{2}$ by $\frac{3}{3}$ .

$\frac{2 y \cdot 2}{6} + \frac{y \cdot 3}{2 \cdot 3} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.4.4

Multiply $2$ by $3$ .

$\frac{2 y \cdot 2}{6} + \frac{y \cdot 3}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 y \cdot 2 + y \cdot 3}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $y$ out of $2 y \cdot 2 + y \cdot 3$ .

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Step 1.6.1.1.1

Factor $y$ out of $2 y \cdot 2$ .

$\frac{y (2 \cdot 2) + y \cdot 3}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6.1.1.2

Factor $y$ out of $y \cdot 3$ .

$\frac{y (2 \cdot 2) + y (3)}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6.1.1.3

Factor $y$ out of $y (2 \cdot 2) + y (3)$ .

$\frac{y (2 \cdot 2 + 3)}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6.1.2

Multiply $2$ by $2$ .

$\frac{y (4 + 3)}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6.1.3

Add $4$ and $3$ .

$\frac{y \cdot 7}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 1.6.2

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

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y}{3} - (4 + y)$

Step 2

Simplify $\frac{5 y}{3} - (4 + y)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y}{3} - 1 \cdot 4 - y$

Step 2.1.2

Multiply $- 1$ by $4$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y}{3} - 4 - y$

Step 2.2

To write $- y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y}{3} - y \cdot \frac{3}{3} - 4$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $- y$ and $\frac{3}{3}$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y}{3} + \frac{- y \cdot 3}{3} - 4$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{7 y}{6} - \frac{7}{2} < \frac{5 y - y \cdot 3}{3} - 4$

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $y$ out of $5 y - y \cdot 3$ .

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Step 2.4.1.1.1

Factor $y$ out of $5 y$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{y \cdot 5 - y \cdot 3}{3} - 4$

Step 2.4.1.1.2

Factor $y$ out of $- y \cdot 3$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{y \cdot 5 + y (- 1 \cdot 3)}{3} - 4$

Step 2.4.1.1.3

Factor $y$ out of $y \cdot 5 + y (- 1 \cdot 3)$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{y (5 - 1 \cdot 3)}{3} - 4$

Step 2.4.1.2

Multiply $- 1$ by $3$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{y (5 - 3)}{3} - 4$

Step 2.4.1.3

Subtract $3$ from $5$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{y \cdot 2}{3} - 4$

Step 2.4.2

Move $2$ to the left of $y$ .

$\frac{7 y}{6} - \frac{7}{2} < \frac{2 y}{3} - 4$

Step 3

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

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Step 3.1

Subtract $\frac{2 y}{3}$ from both sides of the inequality.

$\frac{7 y}{6} - \frac{7}{2} - \frac{2 y}{3} < - 4$

Step 3.2

To write $- \frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 y}{6} - \frac{2 y}{3} \cdot \frac{2}{2} - \frac{7}{2} < - 4$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{2 y}{3}$ by $\frac{2}{2}$ .

$\frac{7 y}{6} - \frac{2 y \cdot 2}{3 \cdot 2} - \frac{7}{2} < - 4$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{7 y}{6} - \frac{2 y \cdot 2}{6} - \frac{7}{2} < - 4$

Step 3.4

Combine the numerators over the common denominator.

$\frac{7 y - 2 y \cdot 2}{6} - \frac{7}{2} < - 4$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $y$ out of $7 y - 2 y \cdot 2$ .

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Step 3.5.1.1.1

Factor $y$ out of $7 y$ .

$\frac{y \cdot 7 - 2 y \cdot 2}{6} - \frac{7}{2} < - 4$

Step 3.5.1.1.2

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

$\frac{y \cdot 7 + y (- 2 \cdot 2)}{6} - \frac{7}{2} < - 4$

Step 3.5.1.1.3

Factor $y$ out of $y \cdot 7 + y (- 2 \cdot 2)$ .

$\frac{y (7 - 2 \cdot 2)}{6} - \frac{7}{2} < - 4$

Step 3.5.1.2

Multiply $- 2$ by $2$ .

$\frac{y (7 - 4)}{6} - \frac{7}{2} < - 4$

Step 3.5.1.3

Subtract $4$ from $7$ .

$\frac{y \cdot 3}{6} - \frac{7}{2} < - 4$

Step 3.5.2

Cancel the common factor of $3$ and $6$ .

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Step 3.5.2.1

Factor $3$ out of $y \cdot 3$ .

$\frac{3 \cdot y}{6} - \frac{7}{2} < - 4$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot y}{3 \cdot 2} - \frac{7}{2} < - 4$

Step 3.5.2.2.2

Cancel the common factor.

$\frac{3 \cdot y}{3 \cdot 2} - \frac{7}{2} < - 4$

Step 3.5.2.2.3

Rewrite the expression.

$\frac{y}{2} - \frac{7}{2} < - 4$

Step 4

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

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Step 4.1

Add $\frac{7}{2}$ to both sides of the inequality.

$\frac{y}{2} < - 4 + \frac{7}{2}$

Step 4.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{2} < - 4 \cdot \frac{2}{2} + \frac{7}{2}$

Step 4.3

Combine $- 4$ and $\frac{2}{2}$ .

$\frac{y}{2} < \frac{- 4 \cdot 2}{2} + \frac{7}{2}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{y}{2} < \frac{- 4 \cdot 2 + 7}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 4$ by $2$ .

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

Step 4.5.2

Add $- 8$ and $7$ .

$\frac{y}{2} < \frac{- 1}{2}$

Step 4.6

Move the negative in front of the fraction.

$\frac{y}{2} < - \frac{1}{2}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$y < - 1$

Step 6

Convert the inequality to interval notation.

$(- \infty, - 1)$

Step 7