Algebra Examples

x - y < 6

$x - y < 6$

2 x + y < 6

$2 x + y < 6$

Step 1

Graph

x - y < 6

$x - y < 6$ .

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Step 1.1

Write in

y = m x + b

$y = m x + b$ form.

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Step 1.1.1

Solve for

y

$y$ .

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Step 1.1.1.1

Subtract

x

$x$ from both sides of the inequality.

- y < 6 - x

$- y < 6 - x$

Step 1.1.1.2

Divide each term in

- y < 6 - x

$- y < 6 - x$ by

- 1

$- 1$ and simplify.

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Step 1.1.1.2.1

Divide each term in

- y < 6 - x

$- y < 6 - x$ by

- 1

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

\frac{- y}{- 1} > \frac{6}{- 1} + \frac{- x}{- 1}

$\frac{- y}{- 1} > \frac{6}{- 1} + \frac{- x}{- 1}$

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} > \frac{6}{- 1} + \frac{- x}{- 1}

$\frac{y}{1} > \frac{6}{- 1} + \frac{- x}{- 1}$

Step 1.1.1.2.2.2

Divide

y

$y$ by

1

$1$ .

y > \frac{6}{- 1} + \frac{- x}{- 1}

$y > \frac{6}{- 1} + \frac{- x}{- 1}$

y > \frac{6}{- 1} + \frac{- x}{- 1}

$y > \frac{6}{- 1} + \frac{- x}{- 1}$

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

6

$6$ by

- 1

$- 1$ .

y > - 6 + \frac{- x}{- 1}

$y > - 6 + \frac{- x}{- 1}$

Step 1.1.1.2.3.1.2

Dividing two negative values results in a positive value.

y > - 6 + \frac{x}{1}

$y > - 6 + \frac{x}{1}$

Step 1.1.1.2.3.1.3

Divide

x

$x$ by

1

$1$ .

y > - 6 + x

$y > - 6 + x$

y > - 6 + x

$y > - 6 + x$

y > - 6 + x

$y > - 6 + x$

y > - 6 + x

$y > - 6 + x$

y > - 6 + x

$y > - 6 + x$

Step 1.1.2

Rearrange terms.

y > x - 6

$y > x - 6$

y > x - 6

$y > x - 6$

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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.2.2

Find the values of

m

$m$ and

b

$b$ using the form

y = m x + b

$y = m x + b$ .

m = 1

$m = 1$

b = - 6

$b = - 6$

Step 1.2.3

The slope of the line is the value of

m

$m$ , and the y-intercept is the value of

b

$b$ .

Slope:

1

$1$

y-intercept:

(0, - 6)

$(0, - 6)$

Slope:

1

$1$

y-intercept:

(0, - 6)

$(0, - 6)$

Step 1.3

Graph a dashed line, then shade the area above the boundary line since

y

$y$ is greater than

x - 6

$x - 6$ .

y > x - 6

$y > x - 6$

y > x - 6

$y > x - 6$

Step 2

Graph

2 x + y < 6

$2 x + y < 6$ .

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Step 2.1

Write in

y = m x + b

$y = m x + b$ form.

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Step 2.1.1

Subtract

2 x

$2 x$ from both sides of the inequality.

y < 6 - 2 x

$y < 6 - 2 x$

Step 2.1.2

Rearrange terms.

y < - 2 x + 6

$y < - 2 x + 6$

y < - 2 x + 6

$y < - 2 x + 6$

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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.2.2

Find the values of

m

$m$ and

b

$b$ using the form

$y = m x + b$ .

$m = - 2$

$b = 6$

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:

$(0, 6)$

Slope:

$- 2$

y-intercept:

$(0, 6)$

Step 2.3

Graph a dashed line, then shade the area below the boundary line since

$y$ is less than

$- 2 x + 6$ .

$y < - 2 x + 6$

Step 3

Plot each graph on the same coordinate system.

$x - y < 6$

$2 x + y < 6$

Step 4