Algebra Examples

$2 x + 3 y \geq 12$ $8 x - 4 y > 1$ $x < 4$

Step 1

Graph $2 x + 3 y \geq 12$ .

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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 $2 x$ from both sides of the inequality.

$3 y \geq 12 - 2 x$

Step 1.1.1.2

Divide each term in $3 y \geq 12 - 2 x$ by $3$ and simplify.

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Step 1.1.1.2.1

Divide each term in $3 y \geq 12 - 2 x$ by $3$ .

$\frac{3 y}{3} \geq \frac{12}{3} + \frac{- 2 x}{3}$

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 $3$ .

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Step 1.1.1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \geq \frac{12}{3} + \frac{- 2 x}{3}$

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{12}{3} + \frac{- 2 x}{3}$

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 $12$ by $3$ .

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

Step 1.1.1.2.3.1.2

Move the negative in front of the fraction.

$y \geq 4 - \frac{2 x}{3}$

Step 1.1.2

Rearrange terms.

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

Step 1.1.3

Reorder terms.

$y \geq - (\frac{2}{3} x) + 4$

Step 1.1.4

Remove parentheses.

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

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 $y = m x + b$ .

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

$b = 4$

Step 1.2.2

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

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

y-intercept: $(0, 4)$

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

y-intercept: $(0, 4)$

Step 1.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $- \frac{2}{3} x + 4$ .

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

Step 2

Graph $8 x - 4 y > 1$ .

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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 $8 x$ from both sides of the inequality.

$- 4 y > 1 - 8 x$

Step 2.1.1.2

Divide each term in $- 4 y > 1 - 8 x$ by $- 4$ and simplify.

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Step 2.1.1.2.1

Divide each term in $- 4 y > 1 - 8 x$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 y}{- 4} < \frac{1}{- 4} + \frac{- 8 x}{- 4}$

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 $- 4$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} < \frac{1}{- 4} + \frac{- 8 x}{- 4}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{1}{- 4} + \frac{- 8 x}{- 4}$

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

Move the negative in front of the fraction.

$y < - \frac{1}{4} + \frac{- 8 x}{- 4}$

Step 2.1.1.2.3.1.2

Cancel the common factor of $- 8$ and $- 4$ .

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Step 2.1.1.2.3.1.2.1

Factor $- 4$ out of $- 8 x$ .

$y < - \frac{1}{4} + \frac{- 4 (2 x)}{- 4}$

Step 2.1.1.2.3.1.2.2

Cancel the common factors.

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Step 2.1.1.2.3.1.2.2.1

Factor $- 4$ out of $- 4$ .

$y < - \frac{1}{4} + \frac{- 4 (2 x)}{- 4 (1)}$

Step 2.1.1.2.3.1.2.2.2

Cancel the common factor.

$y < - \frac{1}{4} + \frac{- 4 (2 x)}{- 4 \cdot 1}$

Step 2.1.1.2.3.1.2.2.3

Rewrite the expression.

$y < - \frac{1}{4} + \frac{2 x}{1}$

Step 2.1.1.2.3.1.2.2.4

Divide $2 x$ by $1$ .

$y < - \frac{1}{4} + 2 x$

Step 2.1.2

Rearrange terms.

$y < 2 x - \frac{1}{4}$

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 $y = m x + b$ .

$m = 2$

$b = - \frac{1}{4}$

Step 2.2.3

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

Slope: $2$

y-intercept: $(0, - \frac{1}{4})$

Slope: $2$

y-intercept: $(0, - \frac{1}{4})$

Step 2.3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $2 x - \frac{1}{4}$ .

$y < 2 x - \frac{1}{4}$

Step 3

Graph $x < 4$ .

$x < 4$

Step 4

Plot each graph on the same coordinate system.

$2 x + 3 y \geq 12$

$8 x - 4 y > 1$

$x < 4$

Step 5