Algebra Examples

$\frac{2 x}{x + 7} - \frac{x}{x + 3} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1

Simplify $\frac{2 x}{x + 7} - \frac{x}{x + 3}$ .

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Step 1.1

To write $\frac{2 x}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{2 x}{x + 7} \cdot \frac{x + 3}{x + 3} - \frac{x}{x + 3} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.2

To write $- \frac{x}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{2 x}{x + 7} \cdot \frac{x + 3}{x + 3} - \frac{x}{x + 3} \cdot \frac{x + 7}{x + 7} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.3

Write each expression with a common denominator of $(x + 7) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 x}{x + 7}$ by $\frac{x + 3}{x + 3}$ .

$\frac{2 x (x + 3)}{(x + 7) (x + 3)} - \frac{x}{x + 3} \cdot \frac{x + 7}{x + 7} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.3.2

Multiply $\frac{x}{x + 3}$ by $\frac{x + 7}{x + 7}$ .

$\frac{2 x (x + 3)}{(x + 7) (x + 3)} - \frac{x (x + 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.3.3

Reorder the factors of $(x + 7) (x + 3)$ .

$\frac{2 x (x + 3)}{(x + 3) (x + 7)} - \frac{x (x + 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2 x (x + 3) - x (x + 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $x$ out of $2 x (x + 3) - x (x + 7)$ .

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Step 1.5.1.1

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

$\frac{x (2 (x + 3)) - x (x + 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.1.2

Factor $x$ out of $- x (x + 7)$ .

$\frac{x (2 (x + 3)) + x (- (x + 7))}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.1.3

Factor $x$ out of $x (2 (x + 3)) + x (- (x + 7))$ .

$\frac{x (2 (x + 3) - (x + 7))}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.2

Apply the distributive property.

$\frac{x (2 x + 2 \cdot 3 - (x + 7))}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.3

Multiply $2$ by $3$ .

$\frac{x (2 x + 6 - (x + 7))}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.4

Apply the distributive property.

$\frac{x (2 x + 6 - x - 1 \cdot 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.5

Multiply $- 1$ by $7$ .

$\frac{x (2 x + 6 - x - 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.6

Subtract $x$ from $2 x$ .

$\frac{x (x + 6 - 7)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 1.5.7

Subtract $7$ from $6$ .

$\frac{x (x - 1)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 10 + 21}$

Step 2

Simplify $1 + \frac{1}{x^{2} + 10 + 21}$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

Add $10$ and $21$ .

$\frac{x (x - 1)}{(x + 3) (x + 7)} = 1 + \frac{1}{x^{2} + 31}$

Step 2.1.2

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

$\frac{x (x - 1)}{(x + 3) (x + 7)} = \frac{x^{2} + 31}{x^{2} + 31} + \frac{1}{x^{2} + 31}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{x (x - 1)}{(x + 3) (x + 7)} = \frac{x^{2} + 31 + 1}{x^{2} + 31}$

Step 2.3

Add $31$ and $1$ .

$\frac{x (x - 1)}{(x + 3) (x + 7)} = \frac{x^{2} + 32}{x^{2} + 31}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.9234097$

Step 4