Pre-Algebra Examples

$- \frac{1}{2} y + \frac{1}{6} x > 1$

Step 1

Write in $y = m x + b$ form.

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Step 1.1

Solve for $y$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

$- \frac{y}{2} + \frac{1}{6} x > 1$

Step 1.1.1.2

Combine $\frac{1}{6}$ and $x$ .

$- \frac{y}{2} + \frac{x}{6} > 1$

Step 1.1.2

Subtract $\frac{x}{6}$ from both sides of the inequality.

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

Step 1.1.3

Divide each term in $- \frac{y}{2} > 1 - \frac{x}{6}$ by $- 1$ and simplify.

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Step 1.1.3.1

Divide each term in $- \frac{y}{2} > 1 - \frac{x}{6}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.3.2.2

Divide $\frac{y}{2}$ by $1$ .

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Divide $1$ by $- 1$ .

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

Step 1.1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.1.3.3.1.3

Divide $\frac{x}{6}$ by $1$ .

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

Step 1.1.4

Multiply both sides by $2$ .

$\frac{y}{2} \cdot 2 < (- 1 + \frac{x}{6}) \cdot 2$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Simplify the left side.

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Step 1.1.5.1.1

Cancel the common factor of $2$ .

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Step 1.1.5.1.1.1

Cancel the common factor.

$\frac{y}{2} \cdot 2 < (- 1 + \frac{x}{6}) \cdot 2$

Step 1.1.5.1.1.2

Rewrite the expression.

$y < (- 1 + \frac{x}{6}) \cdot 2$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify $(- 1 + \frac{x}{6}) \cdot 2$ .

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Step 1.1.5.2.1.1

Apply the distributive property.

$y < - 1 \cdot 2 + \frac{x}{6} \cdot 2$

Step 1.1.5.2.1.2

Multiply $- 1$ by $2$ .

$y < - 2 + \frac{x}{6} \cdot 2$

Step 1.1.5.2.1.3

Cancel the common factor of $2$ .

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Step 1.1.5.2.1.3.1

Factor $2$ out of $6$ .

$y < - 2 + \frac{x}{2 (3)} \cdot 2$

Step 1.1.5.2.1.3.2

Cancel the common factor.

$y < - 2 + \frac{x}{2 \cdot 3} \cdot 2$

Step 1.1.5.2.1.3.3

Rewrite the expression.

$y < - 2 + \frac{x}{3}$

Step 1.1.5.2.1.4

Reorder $- 2$ and $\frac{x}{3}$ .

$y < \frac{x}{3} - 2$

Step 1.2

Reorder terms.

$y < \frac{1}{3} x - 2$

Step 2

Use the slope-intercept form to find the slope and y-intercept.

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Step 2.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{1}{3}$

$b = - 2$

Step 2.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{1}{3}$

y-intercept: $(0, - 2)$

Slope: $\frac{1}{3}$

y-intercept: $(0, - 2)$

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $\frac{1}{3} x - 2$ .

$y < \frac{1}{3} x - 2$

Step 4