Algebra Examples

$3 + \frac{x - 2}{x - 3} \leq 4$

Step 1

Subtract

$4$ from both sides of the inequality.

$3 + \frac{x - 2}{x - 3} - 4 \leq 0$

Step 2

Simplify

$3 + \frac{x - 2}{x - 3} - 4$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Write

$3$ as a fraction with denominator

$1$ .

$\frac{3}{1} + \frac{x - 2}{x - 3} - 4 \leq 0$

Step 2.1.2

Multiply

$\frac{3}{1}$ by

$\frac{x - 3}{x - 3}$ .

$\frac{3}{1} \cdot \frac{x - 3}{x - 3} + \frac{x - 2}{x - 3} - 4 \leq 0$

Step 2.1.3

Multiply

$\frac{3}{1}$ by

$\frac{x - 3}{x - 3}$ .

$\frac{3 (x - 3)}{x - 3} + \frac{x - 2}{x - 3} - 4 \leq 0$

Step 2.1.4

Write

$- 4$ as a fraction with denominator

$1$ .

$\frac{3 (x - 3)}{x - 3} + \frac{x - 2}{x - 3} + \frac{- 4}{1} \leq 0$

Step 2.1.5

Multiply

$\frac{- 4}{1}$ by

$\frac{x - 3}{x - 3}$ .

$\frac{3 (x - 3)}{x - 3} + \frac{x - 2}{x - 3} + \frac{- 4}{1} \cdot \frac{x - 3}{x - 3} \leq 0$

Step 2.1.6

Multiply

$\frac{- 4}{1}$ by

$\frac{x - 3}{x - 3}$ .

$\frac{3 (x - 3)}{x - 3} + \frac{x - 2}{x - 3} + \frac{- 4 (x - 3)}{x - 3} \leq 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{3 (x - 3) + x - 2 - 4 (x - 3)}{x - 3} \leq 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 3 + x - 2 - 4 (x - 3)}{x - 3} \leq 0$

Step 2.3.2

Multiply

$3$ by

$- 3$ .

$\frac{3 x - 9 + x - 2 - 4 (x - 3)}{x - 3} \leq 0$

Step 2.3.3

Apply the distributive property.

$\frac{3 x - 9 + x - 2 - 4 x - 4 \cdot - 3}{x - 3} \leq 0$

Step 2.3.4

Multiply

$- 4$ by

$- 3$ .

$\frac{3 x - 9 + x - 2 - 4 x + 12}{x - 3} \leq 0$

Step 2.4

Add

$3 x$ and

$x$ .

$\frac{4 x - 9 - 2 - 4 x + 12}{x - 3} \leq 0$

Step 2.5

Combine the opposite terms in

$4 x - 9 - 2 - 4 x + 12$ .

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Step 2.5.1

Subtract

$4 x$ from

$4 x$ .

$\frac{0 - 9 - 2 + 12}{x - 3} \leq 0$

Step 2.5.2

Subtract

$9$ from

$0$ .

$\frac{- 9 - 2 + 12}{x - 3} \leq 0$

Step 2.6

Subtract

$2$ from

$- 9$ .

$\frac{- 11 + 12}{x - 3} \leq 0$

Step 2.7

Add

$- 11$ and

$12$ .

$\frac{1}{x - 3} \leq 0$

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

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

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Step 5.1

Set the denominator in

$\frac{1}{x - 3}$ equal to

$0$ to find where the expression is undefined.

$x - 3 = 0$

Step 5.2

Add

$3$ to both sides of the equation.

$x = 3$

Step 5.3

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:

$(- \infty, 3)$

Step 8