Algebra Examples

$y \geq \frac{x^{2} + 3 x + 2}{x + 2}$

Step 1

Simplify $\frac{x^{2} + 3 x + 2}{x + 2}$ .

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Step 1.1

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Cancel the common factor of $x + 2$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $x + 1$ by $1$ .

$y \geq x + 1$

Step 2

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

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Step 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

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

$m = 1$

$b = 1$

Step 2.3

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

Slope: $1$

y-intercept: $(0, 1)$

Slope: $1$

y-intercept: $(0, 1)$

Step 3

Graph a solid line, then shade the area above the boundary line since $y$ is greater than $x + 1$ .

$y \geq x + 1$

Step 4