Algebra Examples

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

Step 1

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

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Step 1.1

Solve for $y$ .

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Step 1.1.1

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

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

Step 1.1.2

Subtract $\frac{2 x}{3}$ from both sides of the inequality.

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

Step 1.1.3

Divide each term in $- 2 y \geq 1 - \frac{2 x}{3}$ by $- 2$ and simplify.

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Step 1.1.3.1

Divide each term in $- 2 y \geq 1 - \frac{2 x}{3}$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.3.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.3.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.3.1.3.1

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

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

Step 1.1.3.3.1.3.2

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

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

Step 1.1.3.3.1.3.3

Factor $2$ out of $- 2$ .

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

Step 1.1.3.3.1.3.4

Cancel the common factor.

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

Step 1.1.3.3.1.3.5

Rewrite the expression.

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

Step 1.1.3.3.1.4

Multiply $\frac{- x}{3}$ by $\frac{1}{- 1}$ .

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

Step 1.1.3.3.1.5

Multiply $3$ by $- 1$ .

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

Step 1.1.3.3.1.6

Dividing two negative values results in a positive value.

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

Step 1.2

Rearrange terms.

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

Step 1.3

Reorder terms.

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

Step 2

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

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Step 2.1

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

$m = \frac{1}{3}$

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

Step 2.2

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

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{1}{2})$

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{1}{2})$

Step 3

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

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

Step 4