Pre-Algebra Examples

$4 x - 2 y > - 15$ , $- x + 3 y > 7$

Step 1

Solve $4 x - 2 y > - 15$ for $y$ .

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Step 1.1

Subtract $4 x$ from both sides of the inequality.

$- 2 y > - 15 - 4 x$

Step 1.2

Divide each term in $- 2 y > - 15 - 4 x$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 y > - 15 - 4 x$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} < \frac{- 15}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} < \frac{- 15}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{- 15}{- 2} + \frac{- 4 x}{- 2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y < \frac{15}{2} + \frac{- 4 x}{- 2}$

Step 1.2.3.1.2

Cancel the common factor of $- 4$ and $- 2$ .

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Step 1.2.3.1.2.1

Factor $- 2$ out of $- 4 x$ .

$y < \frac{15}{2} + \frac{- 2 (2 x)}{- 2}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $- 2$ out of $- 2$ .

$y < \frac{15}{2} + \frac{- 2 (2 x)}{- 2 (1)}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y < \frac{15}{2} + \frac{- 2 (2 x)}{- 2 \cdot 1}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y < \frac{15}{2} + \frac{2 x}{1}$

Step 1.2.3.1.2.2.4

Divide $2 x$ by $1$ .

$y < \frac{15}{2} + 2 x$

Step 2

Solve $- x + 3 y > 7$ for $y$ .

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Step 2.1

Add $x$ to both sides of the inequality.

$3 y > 7 + x$

Step 2.2

Divide each term in $3 y > 7 + x$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 y > 7 + x$ by $3$ .

$\frac{3 y}{3} > \frac{7}{3} + \frac{x}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} > \frac{7}{3} + \frac{x}{3}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{7}{3} + \frac{x}{3}$

Step 3

Find the intersection of $y < \frac{15}{2} + 2 x$ and $y > \frac{7}{3} + \frac{x}{3}$ .

$\frac{7}{3} + \frac{x}{3} < y < \frac{15}{2} + 2 x$

Step 4