Algebra Examples

$\frac{4 x}{2 x + 3} > 2$

Step 1

Subtract $2$ from both sides of the inequality.

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

Step 2

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

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Step 2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2 x + 3}{2 x + 3}$ .

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

Step 2.2

Combine $- 2$ and $\frac{2 x + 3}{2 x + 3}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $2$ out of $4 x - 2 (2 x + 3)$ .

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Step 2.4.1.1

Factor $2$ out of $4 x$ .

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

Step 2.4.1.2

Factor $2$ out of $- 2 (2 x + 3)$ .

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

Step 2.4.1.3

Factor $2$ out of $2 (2 x) + 2 (- (2 x + 3))$ .

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Multiply $2$ by $- 1$ .

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

Step 2.4.4

Multiply $- 1$ by $3$ .

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

Step 2.4.5

Subtract $2 x$ from $2 x$ .

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

Step 2.4.6

Subtract $3$ from $0$ .

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

Step 2.5

Multiply $2$ by $- 3$ .

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

Step 2.6

Move the negative in front of the fraction.

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

Step 3

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

$2 x + 3 = 0$

Step 4

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 5

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 6

Find the domain of $- \frac{6}{2 x + 3}$ .

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Step 6.1

Set the denominator in $\frac{6}{2 x + 3}$ equal to $0$ to find where the expression is undefined.

$2 x + 3 = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 6.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 6.3

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

$(- \infty, - \frac{3}{2}) \cup (- \frac{3}{2}, \infty)$

Step 7

The solution consists of all of the true intervals.

$x < - \frac{3}{2}$

Step 8

Convert the inequality to interval notation.

$(- \infty, - \frac{3}{2})$

Step 9