Solve the Inequality for x 1+(x+2)/(x-3)>=2

$1 + \frac{x + 2}{x - 3} \geq 2$

Step 1

Subtract $2$ from both sides of the inequality.

$1 + \frac{x + 2}{x - 3} - 2 \geq 0$

Step 2

Simplify $1 + \frac{x + 2}{x - 3} - 2$ .

Find the common denominator.

Write $1$ as a fraction with denominator $1$ .

$\frac{1}{1} + \frac{x + 2}{x - 3} - 2 \geq 0$

Multiply $\frac{1}{1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{1}{1} \cdot \frac{x - 3}{x - 3} + \frac{x + 2}{x - 3} - 2 \geq 0$

Multiply $\frac{1}{1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x - 3}{x - 3} + \frac{x + 2}{x - 3} - 2 \geq 0$

Write $- 2$ as a fraction with denominator $1$ .

$\frac{x - 3}{x - 3} + \frac{x + 2}{x - 3} + \frac{- 2}{1} \geq 0$

Multiply $\frac{- 2}{1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x - 3}{x - 3} + \frac{x + 2}{x - 3} + \frac{- 2}{1} \cdot \frac{x - 3}{x - 3} \geq 0$

Multiply $\frac{- 2}{1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x - 3}{x - 3} + \frac{x + 2}{x - 3} + \frac{- 2 (x - 3)}{x - 3} \geq 0$

Combine the numerators over the common denominator.

$\frac{x - 3 + x + 2 - 2 (x - 3)}{x - 3} \geq 0$

Simplify each term.

Apply the distributive property.

$\frac{x - 3 + x + 2 - 2 x - 2 \cdot - 3}{x - 3} \geq 0$

Multiply $- 2$ by $- 3$ .

$\frac{x - 3 + x + 2 - 2 x + 6}{x - 3} \geq 0$

Add $x$ and $x$ .

$\frac{2 x - 3 + 2 - 2 x + 6}{x - 3} \geq 0$

Combine the opposite terms in $2 x - 3 + 2 - 2 x + 6$ .

Subtract $2 x$ from $2 x$ .

$\frac{0 - 3 + 2 + 6}{x - 3} \geq 0$

Subtract $3$ from $0$ .

$\frac{- 3 + 2 + 6}{x - 3} \geq 0$

Add $- 3$ and $2$ .

$\frac{- 1 + 6}{x - 3} \geq 0$

Add $- 1$ and $6$ .

$\frac{5}{x - 3} \geq 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x - 3 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Find the domain of $\frac{5}{x - 3}$ .

Set the denominator in $\frac{5}{x - 3}$ equal to $0$ to find where the expression is undefined.

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 3) \cup (3, \infty)$

Step 6

The solution consists of all of the true intervals.

$x > 3$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > 3$

Interval Notation:

$(3, \infty)$

Step 8

Solve the Inequality for x 1+(x+2)/(x-3)>=2

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