Algebra Examples

$x + 2 y \leq 3$ $x + y \geq 4$ $3 x - 2 y \geq 4$

Step 1

Graph $x + 2 y \leq 3$ .

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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.

$2 y \leq 3 - x$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

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.

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

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.1.2.3

Simplify the right side.

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Step 1.1.1.2.3.1

Move the negative in front of the fraction.

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

Step 1.1.2

Rearrange terms.

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

Step 1.1.3

Reorder terms.

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

Step 1.1.4

Remove parentheses.

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

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 $y = m x + b$ .

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

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

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

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

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

Step 1.3

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

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

Step 2

Graph $x + y \geq 4$ .

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Step 2.1

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

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Step 2.1.1

Subtract $x$ from both sides of the inequality.

$y \geq 4 - x$

Step 2.1.2

Rearrange terms.

$y \geq - x + 4$

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 $y = m x + b$ .

$m = - 1$

$b = 4$

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: $(0, 4)$

Slope: $- 1$

y-intercept: $(0, 4)$

Step 2.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- x + 4$ .

$y \geq - x + 4$

Step 3

Graph $3 x - 2 y \geq 4$ .

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Step 3.1

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

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Step 3.1.1

Solve for $y$ .

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Step 3.1.1.1

Subtract $3 x$ from both sides of the inequality.

$- 2 y \geq 4 - 3 x$

Step 3.1.1.2

Divide each term in $- 2 y \geq 4 - 3 x$ by $- 2$ and simplify.

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Step 3.1.1.2.1

Divide each term in $- 2 y \geq 4 - 3 x$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 y}{- 2} \leq \frac{4}{- 2} + \frac{- 3 x}{- 2}$

Step 3.1.1.2.2

Simplify the left side.

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Step 3.1.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} \leq \frac{4}{- 2} + \frac{- 3 x}{- 2}$

Step 3.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{4}{- 2} + \frac{- 3 x}{- 2}$

Step 3.1.1.2.3

Simplify the right side.

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Step 3.1.1.2.3.1

Simplify each term.

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Step 3.1.1.2.3.1.1

Divide $4$ by $- 2$ .

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

Step 3.1.1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 3.1.2

Rearrange terms.

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

Step 3.1.3

Reorder terms.

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

Step 3.2

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

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Step 3.2.1

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

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

$b = - 2$

Step 3.2.2

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

Slope: $\frac{3}{2}$

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

Slope: $\frac{3}{2}$

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

Step 3.3

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

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

Step 4

Plot each graph on the same coordinate system.

$x + 2 y \leq 3$

$x + y \geq 4$

$3 x - 2 y \geq 4$

Step 5