Calculus Examples

$5 y < 7 - 2 y < 8 + y$

Step 1

Move all terms not containing $y$ from the center section of the inequality.

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Step 1.1

Subtract $7$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$5 y - 7 < - 2 y < 8 + y - 7$

Step 1.2

Subtract $7$ from $8$ .

$5 y - 7 < - 2 y < y + 1$

Step 2

Divide each term in the inequality by $- 2$ .

$\frac{5 y}{- 2} + \frac{- 7}{- 2} > \frac{- 2 y}{- 2} > \frac{y}{- 2} + \frac{1}{- 2}$

Step 3

Simplify each term.

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Step 3.1

Move the negative in front of the fraction.

$- \frac{5 y}{2} + \frac{- 7}{- 2} > \frac{- 2 y}{- 2} > \frac{y}{- 2} + \frac{1}{- 2}$

Step 3.2

Dividing two negative values results in a positive value.

$- \frac{5 y}{2} + \frac{7}{2} > \frac{- 2 y}{- 2} > \frac{y}{- 2} + \frac{1}{- 2}$

Step 4

Cancel the common factor of $- 2$ .

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Step 4.1

Cancel the common factor.

$- \frac{5 y}{2} + \frac{7}{2} > \frac{- 2 y}{- 2} > \frac{y}{- 2} + \frac{1}{- 2}$

Step 4.2

Divide $y$ by $1$ .

$- \frac{5 y}{2} + \frac{7}{2} > y > \frac{y}{- 2} + \frac{1}{- 2}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{5 y}{2} + \frac{7}{2} > y > - \frac{y}{2} + \frac{1}{- 2}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{5 y}{2} + \frac{7}{2} > y > - \frac{y}{2} - \frac{1}{2}$

Step 6

Convert the inequality to interval notation.

$(- \frac{5 y}{2} + \frac{7}{2}, - \frac{y}{2} - \frac{1}{2})$

Step 7