Precalculus Examples

$\frac{x + 4}{2} + \frac{x - 1}{2} = \frac{x + 4}{2 x}$

Step 1

Subtract $\frac{x + 4}{2 x}$ from both sides of the equation.

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

Step 2

Simplify $\frac{x + 4}{2} + \frac{x - 1}{2} - \frac{x + 4}{2 x}$ .

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $x$ and $x$ .

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

Step 2.3

Subtract $1$ from $4$ .

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

Step 2.4

Move the negative in front of the fraction.

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

Step 2.5

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

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

Step 2.6

Multiply $\frac{2 x + 3}{2}$ by $\frac{x}{x}$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.8.2.1

Move $x$ .

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

Step 2.8.2.2

Multiply $x$ by $x$ .

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

Step 2.8.3

Apply the distributive property.

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

Step 2.8.4

Multiply $- 1$ by $4$ .

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

Step 2.8.5

Subtract $x$ from $3 x$ .

$\frac{2 x^{2} + 2 x - 4}{2 x} = 0$

Step 2.8.6

Rewrite $2 x^{2} + 2 x - 4$ in a factored form.

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Step 2.8.6.1

Factor $2$ out of $2 x^{2} + 2 x - 4$ .

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Step 2.8.6.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) + 2 x - 4}{2 x} = 0$

Step 2.8.6.1.2

Factor $2$ out of $2 x$ .

$\frac{2 (x^{2}) + 2 (x) - 4}{2 x} = 0$

Step 2.8.6.1.3

Factor $2$ out of $- 4$ .

$\frac{2 (x^{2}) + 2 x + 2 \cdot - 2}{2 x} = 0$

Step 2.8.6.1.4

Factor $2$ out of $2 (x^{2}) + 2 x$ .

$\frac{2 (x^{2} + x) + 2 \cdot - 2}{2 x} = 0$

Step 2.8.6.1.5

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 2$ .

$\frac{2 (x^{2} + x - 2)}{2 x} = 0$

Step 2.8.6.2

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

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Step 2.8.6.2.1

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 $1$ .

$- 1, 2$

Step 2.8.6.2.2

Write the factored form using these integers.

$\frac{2 ((x - 1) (x + 2))}{2 x} = 0$

$\frac{2 (x - 1) (x + 2)}{2 x} = 0$

Step 2.9

Cancel the common factor of $2$ .

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Step 2.9.1

Cancel the common factor.

$\frac{2 (x - 1) (x + 2)}{2 x} = 0$

Step 2.9.2

Rewrite the expression.

$\frac{(x - 1) (x + 2)}{x} = 0$

Step 3

Set the numerator equal to zero.

$(x - 1) (x + 2) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 2 = 0$

Step 4.2

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.4

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$