Algebra Examples

y \leq 2 x - 4

$y \leq 2 x - 4$

2 y - x \geq 2

$2 y - x \geq 2$

Step 1

Graph

y \leq 2 x - 4

$y \leq 2 x - 4$ .

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Step 1.1

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

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Step 1.1.1

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 1.1.2

Find the values of

m

$m$ and

b

$b$ using the form

y = m x + b

$y = m x + b$ .

m = 2

$m = 2$

b = - 4

$b = - 4$

Step 1.1.3

The slope of the line is the value of

m

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

b

$b$ .

Slope:

2

$2$

y-intercept:

(0, - 4)

$(0, - 4)$

Slope:

2

$2$

y-intercept:

(0, - 4)

$(0, - 4)$

Step 1.2

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

y

$y$ is less than

2 x - 4

$2 x - 4$ .

y \leq 2 x - 4

$y \leq 2 x - 4$

y \leq 2 x - 4

$y \leq 2 x - 4$

Step 2

Graph

2 y - x \geq 2

$2 y - x \geq 2$ .

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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

Solve for

y

$y$ .

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Step 2.1.1.1

Add

x

$x$ to both sides of the inequality.

2 y \geq 2 + x

$2 y \geq 2 + x$

Step 2.1.1.2

Divide each term in

2 y \geq 2 + x

$2 y \geq 2 + x$ by

2

$2$ and simplify.

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Step 2.1.1.2.1

Divide each term in

2 y \geq 2 + x

$2 y \geq 2 + x$ by

2

$2$ .

\frac{2 y}{2} \geq \frac{2}{2} + \frac{x}{2}

$\frac{2 y}{2} \geq \frac{2}{2} + \frac{x}{2}$

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

$2$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} \geq \frac{2}{2} + \frac{x}{2}$

Step 2.1.1.2.2.1.2

Divide

$y$ by

$1$ .

$y \geq \frac{2}{2} + \frac{x}{2}$

Step 2.1.1.2.3

Simplify the right side.

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Step 2.1.1.2.3.1

Divide

$2$ by

$2$ .

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

Step 2.1.2

Rearrange terms.

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

Step 2.1.3

Reorder terms.

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

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 solid line, then shade the area above the boundary line since

$y$ is greater than

$\frac{1}{2} x + 1$ .

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

Step 3

Plot each graph on the same coordinate system.

$y \leq 2 x - 4$

$2 y - x \geq 2$

Step 4