Algebra Examples

$\frac{1}{2} x + 6 > 2 y$ $\frac{1}{3} y < \frac{2}{3} x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Solve for $y$ .

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Divide each term in $2 y < \frac{1}{2} x + 6$ by $2$ .

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

Step 1.1.1.2.2

Simplify the left side.

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Step 1.1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.1.2.3

Simplify the right side.

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Step 1.1.1.2.3.1

Simplify each term.

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Step 1.1.1.2.3.1.1

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

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

Step 1.1.1.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.1.2.3.1.3

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

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Step 1.1.1.2.3.1.3.1

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

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

Step 1.1.1.2.3.1.3.2

Multiply $2$ by $2$ .

$y < \frac{x}{4} + \frac{6}{2}$

Step 1.1.1.2.3.1.4

Divide $6$ by $2$ .

$y < \frac{x}{4} + 3$

Step 1.1.2

Reorder terms.

$y < \frac{1}{4} x + 3$

Step 1.2

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

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Step 1.2.1

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

$m = \frac{1}{4}$

$b = 3$

Step 1.2.2

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

Slope: $\frac{1}{4}$

y-intercept: $(0, 3)$

Slope: $\frac{1}{4}$

y-intercept: $(0, 3)$

Step 1.3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $\frac{1}{4} x + 3$ .

$y < \frac{1}{4} x + 3$

Step 2

Graph $\frac{1}{3} y < \frac{2}{3} x$ .

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Step 2.1

Multiply each term in $\frac{1}{3} y < \frac{2}{3} x$ by $3$ to eliminate the fractions.

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Step 2.1.1

Multiply each term in $\frac{1}{3} y < \frac{2}{3} x$ by $3$ .

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Combine $\frac{1}{3}$ and $y$ .

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

Step 2.1.2.2

Cancel the common factor of $3$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Combine $\frac{2}{3}$ and $x$ .

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

Step 2.1.3.2

Cancel the common factor of $3$ .

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Step 2.1.3.2.1

Cancel the common factor.

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

Step 2.1.3.2.2

Rewrite the expression.

$y < 2 x$

Step 2.2

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

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Step 2.2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2.2

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

$m = 2$

$b = 0$

Step 2.2.3

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

Slope: $2$

y-intercept: $(0, 0)$

Slope: $2$

y-intercept: $(0, 0)$

Step 2.3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $2 x$ .

$y < 2 x$

Step 3

Plot each graph on the same coordinate system.

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

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

Step 4