Algebra Examples

$2 x - y \leq 5$ $x + 2 y > 2$

Step 1

Graph $2 x - y \leq 5$ .

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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.

$- y \leq 5 - 2 x$

Step 1.1.1.2

Divide each term in $- y \leq 5 - 2 x$ by $- 1$ and simplify.

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Step 1.1.1.2.1

Divide each term in $- y \leq 5 - 2 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.1.1.2.2

Simplify the left side.

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Step 1.1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.1.2.2.2

Divide $y$ by $1$ .

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

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 $5$ by $- 1$ .

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

Step 1.1.1.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$y \geq - 5 - 1 \cdot (- 2 x)$

Step 1.1.1.2.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$y \geq - 5 - (- 2 x)$

Step 1.1.1.2.3.1.4

Multiply $- 2$ by $- 1$ .

$y \geq - 5 + 2 x$

Step 1.1.2

Rearrange terms.

$y \geq 2 x - 5$

Step 1.2

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

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Step 1.2.1

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 1.2.2

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

$m = 2$

$b = - 5$

Step 1.2.3

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

Slope: $2$

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

Slope: $2$

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

Step 1.3

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

$y \geq 2 x - 5$

Step 2

Graph $x + 2 y > 2$ .

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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.

$2 y > 2 - x$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Divide each term in $2 y > 2 - x$ by $2$ .

$\frac{2 y}{2} > \frac{2}{2} + \frac{- x}{2}$

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.

$\frac{2 y}{2} > \frac{2}{2} + \frac{- x}{2}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{2}{2} + \frac{- x}{2}$

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 $2$ by $2$ .

$y > 1 + \frac{- x}{2}$

Step 2.1.1.2.3.1.2

Move the negative in front of the fraction.

$y > 1 - \frac{x}{2}$

Step 2.1.2

Rearrange terms.

$y > - \frac{x}{2} + 1$

Step 2.1.3

Reorder terms.

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

Step 2.1.4

Remove parentheses.

$y > - \frac{1}{2} x + 1$

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 $y = m x + b$ .

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

$b = 1$

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: $(0, 1)$

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

y-intercept: $(0, 1)$

Step 2.3

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

$y > - \frac{1}{2} x + 1$

Step 3

Plot each graph on the same coordinate system.

$2 x - y \leq 5$

$x + 2 y > 2$

Step 4