Algebra Examples

$2 y - x \leq 2$ $y - 3 x \geq - 4$ $y \geq - 1$

Step 1

Graph $2 y - x \leq 2$ .

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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

Add $x$ to both sides of the inequality.

$2 y \leq 2 + x$

Step 1.1.1.2

Divide each term in $2 y \leq 2 + x$ by $2$ and simplify.

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Step 1.1.1.2.1

Divide each term in $2 y \leq 2 + x$ by $2$ .

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

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 $2$ .

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Step 1.1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.1.2.3

Simplify the right side.

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Step 1.1.1.2.3.1

Divide $2$ by $2$ .

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

Step 1.1.2

Rearrange terms.

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

Step 1.1.3

Reorder terms.

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

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 $y = m x + b$ .

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

$b = 1$

Step 1.2.2

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

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

y-intercept: $(0, 1)$

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

y-intercept: $(0, 1)$

Step 1.3

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

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

Step 2

Graph $y - 3 x \geq - 4$ .

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Step 2.1

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

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Step 2.1.1

Add $3 x$ to both sides of the inequality.

$y \geq - 4 + 3 x$

Step 2.1.2

Rearrange terms.

$y \geq 3 x - 4$

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 $y = m x + b$ .

$m = 3$

$b = - 4$

Step 2.2.3

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

Slope: $3$

y-intercept: $(0, - 4)$

Slope: $3$

y-intercept: $(0, - 4)$

Step 2.3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $3 x - 4$ .

$y \geq 3 x - 4$

Step 3

Graph $y \geq - 1$ .

$y \geq - 1$

Step 4

Plot each graph on the same coordinate system.

$2 y - x \leq 2$

$y - 3 x \geq - 4$

$y \geq - 1$

Step 5