Precalculus Examples

$x + y < 7$ , $y < \frac{1}{2} x - 6$ , $3 x + y < 6$

Step 1

Subtract $y$ from both sides of the inequality.

$x < 7 - y$ or $y < \frac{1}{2} x - 6$

Step 2

Simplify the second inequality.

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Step 2.1

Rewrite so $x$ is on the left side of the inequality.

$x < 7 - y$ or $\frac{1}{2} x - 6 > y$

Step 2.2

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

$x < 7 - y$ or $\frac{x}{2} - 6 > y$

Step 2.3

Add $6$ to both sides of the inequality.

$x < 7 - y$ or $\frac{x}{2} > y + 6$

Step 2.4

Multiply both sides by $2$ .

$x < 7 - y$ or $\frac{x}{2} \cdot 2 > (y + 6) \cdot 2$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $2$ .

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Step 2.5.1.1.1

Cancel the common factor.

$x < 7 - y$ or $\frac{x}{2} \cdot 2 > (y + 6) \cdot 2$

Step 2.5.1.1.2

Rewrite the expression.

$x < 7 - y$ or $x > (y + 6) \cdot 2$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $(y + 6) \cdot 2$ .

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Step 2.5.2.1.1

Apply the distributive property.

$x < 7 - y$ or $x > y \cdot 2 + 6 \cdot 2$

Step 2.5.2.1.2

Simplify the expression.

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Step 2.5.2.1.2.1

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

$x < 7 - y$ or $x > 2 \cdot y + 6 \cdot 2$

Step 2.5.2.1.2.2

Multiply $6$ by $2$ .

$x < 7 - y$ or $x > 2 y + 12$

Step 3