Solve the System of Inequalities 5/4-2y<3/2 and 3/5y+3<13/10

$\frac{5}{4} - 2 y < \frac{3}{2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Step 1

Simplify the first inequality.

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

Subtract $\frac{5}{4}$ from both sides of the inequality.

$- 2 y < \frac{3}{2} - \frac{5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

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

$- 2 y < \frac{3}{2} \cdot \frac{2}{2} - \frac{5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$- 2 y < \frac{3 \cdot 2}{2 \cdot 2} - \frac{5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Multiply $2$ by $2$ .

$- 2 y < \frac{3 \cdot 2}{4} - \frac{5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Combine the numerators over the common denominator.

$- 2 y < \frac{3 \cdot 2 - 5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Simplify the numerator.

Multiply $3$ by $2$ .

$- 2 y < \frac{6 - 5}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Subtract $5$ from $6$ .

$- 2 y < \frac{1}{4}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Divide each term in $- 2 y < \frac{1}{4}$ by $- 2$ and simplify.

Divide each term in $- 2 y < \frac{1}{4}$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} > \frac{\frac{1}{4}}{- 2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} > \frac{\frac{1}{4}}{- 2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Divide $y$ by $1$ .

$y > \frac{\frac{1}{4}}{- 2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$y > \frac{1}{4} \cdot \frac{1}{- 2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Move the negative in front of the fraction.

$y > \frac{1}{4} \cdot (- \frac{1}{2})$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Multiply $\frac{1}{4} (- \frac{1}{2})$ .

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$y > - \frac{1}{4 \cdot 2}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Multiply $4$ by $2$ .

$y > - \frac{1}{8}$ and $\frac{3}{5} y + 3 < \frac{13}{10}$

Step 2

Simplify the second inequality.

Combine $\frac{3}{5}$ and $y$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} + 3 < \frac{13}{10}$

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

Subtract $3$ from both sides of the inequality.

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{13}{10} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{13}{10} - 3 \cdot \frac{10}{10}$

Combine $- 3$ and $\frac{10}{10}$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{13}{10} + \frac{- 3 \cdot 10}{10}$

Combine the numerators over the common denominator.

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{13 - 3 \cdot 10}{10}$

Simplify the numerator.

Multiply $- 3$ by $10$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{13 - 30}{10}$

Subtract $30$ from $13$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < \frac{- 17}{10}$

Move the negative in front of the fraction.

$y > - \frac{1}{8}$ and $\frac{3 y}{5} < - \frac{17}{10}$

Multiply both sides by $5$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{5} \cdot 5 < - \frac{17}{10} \cdot 5$

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$y > - \frac{1}{8}$ and $\frac{3 y}{5} \cdot 5 < - \frac{17}{10} \cdot 5$

Rewrite the expression.

$y > - \frac{1}{8}$ and $3 y < - \frac{17}{10} \cdot 5$

Simplify the right side.

Simplify $- \frac{17}{10} \cdot 5$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{17}{10}$ into the numerator.

$y > - \frac{1}{8}$ and $3 y < \frac{- 17}{10} \cdot 5$

Factor $5$ out of $10$ .

$y > - \frac{1}{8}$ and $3 y < \frac{- 17}{5 (2)} \cdot 5$

Cancel the common factor.

$y > - \frac{1}{8}$ and $3 y < \frac{- 17}{5 \cdot 2} \cdot 5$

Rewrite the expression.

$y > - \frac{1}{8}$ and $3 y < \frac{- 17}{2}$

Move the negative in front of the fraction.

$y > - \frac{1}{8}$ and $3 y < - \frac{17}{2}$

Divide each term in $3 y < - \frac{17}{2}$ by $3$ and simplify.

Divide each term in $3 y < - \frac{17}{2}$ by $3$ .

$y > - \frac{1}{8}$ and $\frac{3 y}{3} < \frac{- \frac{17}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$y > - \frac{1}{8}$ and $\frac{3 y}{3} < \frac{- \frac{17}{2}}{3}$

Divide $y$ by $1$ .

$y > - \frac{1}{8}$ and $y < \frac{- \frac{17}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$y > - \frac{1}{8}$ and $y < - \frac{17}{2} \cdot \frac{1}{3}$

Multiply $- \frac{17}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{1}{3}$ by $\frac{17}{2}$ .

$y > - \frac{1}{8}$ and $y < - \frac{17}{3 \cdot 2}$

Multiply $3$ by $2$ .

$y > - \frac{1}{8}$ and $y < - \frac{17}{6}$

Step 3

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

No solution

Solve the System of Inequalities 5/4-2y<3/2 and 3/5y+3<13/10

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