Algebra Examples

$\frac{1}{2} (x + 2) + 3 y < 8$

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

Apply the distributive property.

$\frac{1}{2} x + \frac{1}{2} \cdot 2 + 3 y < 8$

Step 1.1.1.2

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

$\frac{x}{2} + \frac{1}{2} \cdot 2 + 3 y < 8$

Step 1.1.1.3

Cancel the common factor of $2$ .

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Step 1.1.1.3.1

Cancel the common factor.

$\frac{x}{2} + \frac{1}{2} \cdot 2 + 3 y < 8$

Step 1.1.1.3.2

Rewrite the expression.

$\frac{x}{2} + 1 + 3 y < 8$

Step 1.1.2

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

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Step 1.1.2.1

Subtract $\frac{x}{2}$ from both sides of the inequality.

$1 + 3 y < 8 - \frac{x}{2}$

Step 1.1.2.2

Subtract $1$ from both sides of the inequality.

$3 y < 8 - \frac{x}{2} - 1$

Step 1.1.2.3

Subtract $1$ from $8$ .

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

Step 1.1.3

Divide each term in $3 y < - \frac{x}{2} + 7$ by $3$ and simplify.

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Step 1.1.3.1

Divide each term in $3 y < - \frac{x}{2} + 7$ by $3$ .

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $3$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.3.1.2

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

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Step 1.1.3.3.1.2.1

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

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

Step 1.1.3.3.1.2.2

Multiply $3$ by $2$ .

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

Step 1.2

Reorder terms.

$y < - (\frac{1}{6} x) + \frac{7}{3}$

Step 1.3

Remove parentheses.

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

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 $y = m x + b$ .

$m = - \frac{1}{6}$

$b = \frac{7}{3}$

Step 2.2

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

Slope: $- \frac{1}{6}$

y-intercept: $(0, \frac{7}{3})$

Slope: $- \frac{1}{6}$

y-intercept: $(0, \frac{7}{3})$

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $- \frac{1}{6} x + \frac{7}{3}$ .

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

Step 4