Algebra Examples

$y \geq 3 x$ $2 x + y < 3 - y$

Step 1

Graph $y \geq 3 x$ .

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Step 1.1

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

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Step 1.1.1

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 1.1.2

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

$m = 3$

$b = 0$

Step 1.1.3

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

Slope: $3$

y-intercept: $(0, 0)$

Slope: $3$

y-intercept: $(0, 0)$

Step 1.2

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $3 x$ .

$y \geq 3 x$

Step 2

Graph $2 x + y < 3 - y$ .

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Step 2.1

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

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Step 2.1.1

Solve for $y$ .

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Step 2.1.1.1

Move all terms containing $y$ to the left side of the inequality.

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Step 2.1.1.1.1

Add $y$ to both sides of the inequality.

$2 x + y + y < 3$

Step 2.1.1.1.2

Add $y$ and $y$ .

$2 x + 2 y < 3$

Step 2.1.1.2

Subtract $2 x$ from both sides of the inequality.

$2 y < 3 - 2 x$

Step 2.1.1.3

Divide each term in $2 y < 3 - 2 x$ by $2$ and simplify.

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Step 2.1.1.3.1

Divide each term in $2 y < 3 - 2 x$ by $2$ .

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

Step 2.1.1.3.2

Simplify the left side.

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Step 2.1.1.3.2.1

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.1.1

Cancel the common factor.

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

Step 2.1.1.3.2.1.2

Divide $y$ by $1$ .

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

Step 2.1.1.3.3

Simplify the right side.

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Step 2.1.1.3.3.1

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

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Step 2.1.1.3.3.1.1

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

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

Step 2.1.1.3.3.1.2

Cancel the common factors.

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Step 2.1.1.3.3.1.2.1

Factor $2$ out of $2$ .

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

Step 2.1.1.3.3.1.2.2

Cancel the common factor.

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

Step 2.1.1.3.3.1.2.3

Rewrite the expression.

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

Step 2.1.1.3.3.1.2.4

Divide $- x$ by $1$ .

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

Step 2.1.2

Rearrange terms.

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

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 $y = m x + b$ .

$m = - 1$

$b = \frac{3}{2}$

Step 2.2.3

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

Slope: $- 1$

y-intercept: $(0, \frac{3}{2})$

Slope: $- 1$

y-intercept: $(0, \frac{3}{2})$

Step 2.3

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

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

Step 3

Plot each graph on the same coordinate system.

$y \geq 3 x$

$2 x + y < 3 - y$

Step 4