Algebra Examples

$\frac{2 y + 1}{5} - \frac{y - 2}{3} \leq - \frac{1}{15}$

Step 1

Simplify $\frac{2 y + 1}{5} - \frac{y - 2}{3}$ .

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Step 1.1

To write $\frac{2 y + 1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 y + 1}{5} \cdot \frac{3}{3} - \frac{y - 2}{3} \leq - \frac{1}{15}$

Step 1.2

To write $- \frac{y - 2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 y + 1}{5} \cdot \frac{3}{3} - \frac{y - 2}{3} \cdot \frac{5}{5} \leq - \frac{1}{15}$

Step 1.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 y + 1}{5}$ by $\frac{3}{3}$ .

$\frac{(2 y + 1) \cdot 3}{5 \cdot 3} - \frac{y - 2}{3} \cdot \frac{5}{5} \leq - \frac{1}{15}$

Step 1.3.2

Multiply $5$ by $3$ .

$\frac{(2 y + 1) \cdot 3}{15} - \frac{y - 2}{3} \cdot \frac{5}{5} \leq - \frac{1}{15}$

Step 1.3.3

Multiply $\frac{y - 2}{3}$ by $\frac{5}{5}$ .

$\frac{(2 y + 1) \cdot 3}{15} - \frac{(y - 2) \cdot 5}{3 \cdot 5} \leq - \frac{1}{15}$

Step 1.3.4

Multiply $3$ by $5$ .

$\frac{(2 y + 1) \cdot 3}{15} - \frac{(y - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(2 y + 1) \cdot 3 - (y - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{2 y \cdot 3 + 1 \cdot 3 - (y - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.2

Multiply $3$ by $2$ .

$\frac{6 y + 1 \cdot 3 - (y - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.3

Multiply $3$ by $1$ .

$\frac{6 y + 3 - (y - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.4

Apply the distributive property.

$\frac{6 y + 3 + (- y - - 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.5

Multiply $- 1$ by $- 2$ .

$\frac{6 y + 3 + (- y + 2) \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.6

Apply the distributive property.

$\frac{6 y + 3 - y \cdot 5 + 2 \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.7

Multiply $5$ by $- 1$ .

$\frac{6 y + 3 - 5 y + 2 \cdot 5}{15} \leq - \frac{1}{15}$

Step 1.5.8

Multiply $2$ by $5$ .

$\frac{6 y + 3 - 5 y + 10}{15} \leq - \frac{1}{15}$

Step 1.5.9

Subtract $5 y$ from $6 y$ .

$\frac{y + 3 + 10}{15} \leq - \frac{1}{15}$

Step 1.5.10

Add $3$ and $10$ .

$\frac{y + 13}{15} \leq - \frac{1}{15}$

Step 2

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

$y + 13 \leq - 1$

Step 3

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

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Step 3.1

Subtract $13$ from both sides of the inequality.

$y \leq - 1 - 13$

Step 3.2

Subtract $13$ from $- 1$ .

$y \leq - 14$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$y \leq - 14$

Interval Notation:

$(- \infty, - 14]$