Algebra Examples

$x + 2 y \geq 6$ $x + 2 y \geq - 10$

Step 1

Graph $x + 2 y \geq 6$ .

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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

Subtract $x$ from both sides of the inequality.

$2 y \geq 6 - x$

Step 1.1.1.2

Divide each term in $2 y \geq 6 - x$ by $2$ and simplify.

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Step 1.1.1.2.1

Divide each term in $2 y \geq 6 - x$ by $2$ .

$\frac{2 y}{2} \geq \frac{6}{2} + \frac{- x}{2}$

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.

$\frac{2 y}{2} \geq \frac{6}{2} + \frac{- x}{2}$

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{6}{2} + \frac{- x}{2}$

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

Divide $6$ by $2$ .

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

Step 1.1.1.2.3.1.2

Move the negative in front of the fraction.

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

Step 1.1.2

Rearrange terms.

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

Step 1.1.3

Reorder terms.

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

Step 1.1.4

Remove parentheses.

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

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 $y = m x + b$ .

$m = - \frac{1}{2}$

$b = 3$

Step 1.2.2

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

Slope: $- \frac{1}{2}$

y-intercept: $(0, 3)$

Slope: $- \frac{1}{2}$

y-intercept: $(0, 3)$

Step 1.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- \frac{1}{2} x + 3$ .

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

Step 2

Graph $x + 2 y \geq - 10$ .

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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

Subtract $x$ from both sides of the inequality.

$2 y \geq - 10 - x$

Step 2.1.1.2

Divide each term in $2 y \geq - 10 - x$ by $2$ and simplify.

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Step 2.1.1.2.1

Divide each term in $2 y \geq - 10 - x$ by $2$ .

$\frac{2 y}{2} \geq \frac{- 10}{2} + \frac{- x}{2}$

Step 2.1.1.2.2

Simplify the left side.

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Step 2.1.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} \geq \frac{- 10}{2} + \frac{- x}{2}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{- 10}{2} + \frac{- x}{2}$

Step 2.1.1.2.3

Simplify the right side.

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Step 2.1.1.2.3.1

Simplify each term.

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Step 2.1.1.2.3.1.1

Divide $- 10$ by $2$ .

$y \geq - 5 + \frac{- x}{2}$

Step 2.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \geq - 5 - \frac{x}{2}$

Step 2.1.2

Rearrange terms.

$y \geq - \frac{x}{2} - 5$

Step 2.1.3

Reorder terms.

$y \geq - (\frac{1}{2} x) - 5$

Step 2.1.4

Remove parentheses.

$y \geq - \frac{1}{2} x - 5$

Step 2.2

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

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Step 2.2.1

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

$m = - \frac{1}{2}$

$b = - 5$

Step 2.2.2

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

Slope: $- \frac{1}{2}$

y-intercept: $(0, - 5)$

Slope: $- \frac{1}{2}$

y-intercept: $(0, - 5)$

Step 2.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- \frac{1}{2} x - 5$ .

$y \geq - \frac{1}{2} x - 5$

Step 3

Plot each graph on the same coordinate system.

$x + 2 y \geq 6$

$x + 2 y \geq - 10$

Step 4