Pre-Algebra Examples

$\frac{320}{x} + 1 - \frac{420}{x + 4} = 0$

Step 1

Simplify $\frac{320}{x} + 1 - \frac{420}{x + 4}$ .

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Step 1.1

To write $\frac{320}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{320}{x} \cdot \frac{x + 4}{x + 4} - \frac{420}{x + 4} + 1 = 0$

Step 1.2

To write $- \frac{420}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{320}{x} \cdot \frac{x + 4}{x + 4} - \frac{420}{x + 4} \cdot \frac{x}{x} + 1 = 0$

Step 1.3

Write each expression with a common denominator of $x (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{320}{x}$ by $\frac{x + 4}{x + 4}$ .

$\frac{320 (x + 4)}{x (x + 4)} - \frac{420}{x + 4} \cdot \frac{x}{x} + 1 = 0$

Step 1.3.2

Multiply $\frac{420}{x + 4}$ by $\frac{x}{x}$ .

$\frac{320 (x + 4)}{x (x + 4)} - \frac{420 x}{(x + 4) x} + 1 = 0$

Step 1.3.3

Reorder the factors of $(x + 4) x$ .

$\frac{320 (x + 4)}{x (x + 4)} - \frac{420 x}{x (x + 4)} + 1 = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{320 (x + 4) - 420 x}{x (x + 4)} + 1 = 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $20$ out of $320 (x + 4) - 420 x$ .

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Step 1.5.1.1

Factor $20$ out of $320 (x + 4)$ .

$\frac{20 (16 (x + 4)) - 420 x}{x (x + 4)} + 1 = 0$

Step 1.5.1.2

Factor $20$ out of $- 420 x$ .

$\frac{20 (16 (x + 4)) + 20 (- 21 x)}{x (x + 4)} + 1 = 0$

Step 1.5.1.3

Factor $20$ out of $20 (16 (x + 4)) + 20 (- 21 x)$ .

$\frac{20 (16 (x + 4) - 21 x)}{x (x + 4)} + 1 = 0$

Step 1.5.2

Apply the distributive property.

$\frac{20 (16 x + 16 \cdot 4 - 21 x)}{x (x + 4)} + 1 = 0$

Step 1.5.3

Multiply $16$ by $4$ .

$\frac{20 (16 x + 64 - 21 x)}{x (x + 4)} + 1 = 0$

Step 1.5.4

Subtract $21 x$ from $16 x$ .

$\frac{20 (- 5 x + 64)}{x (x + 4)} + 1 = 0$

Step 1.6

Combine into one fraction.

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Step 1.6.1

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

$\frac{20 (- 5 x + 64)}{x (x + 4)} + \frac{x (x + 4)}{x (x + 4)} = 0$

Step 1.6.2

Combine the numerators over the common denominator.

$\frac{20 (- 5 x + 64) + x (x + 4)}{x (x + 4)} = 0$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Apply the distributive property.

$\frac{20 (- 5 x) + 20 \cdot 64 + x (x + 4)}{x (x + 4)} = 0$

Step 1.7.2

Multiply $- 5$ by $20$ .

$\frac{- 100 x + 20 \cdot 64 + x (x + 4)}{x (x + 4)} = 0$

Step 1.7.3

Multiply $20$ by $64$ .

$\frac{- 100 x + 1280 + x (x + 4)}{x (x + 4)} = 0$

Step 1.7.4

Apply the distributive property.

$\frac{- 100 x + 1280 + x \cdot x + x \cdot 4}{x (x + 4)} = 0$

Step 1.7.5

Multiply $x$ by $x$ .

$\frac{- 100 x + 1280 + x^{2} + x \cdot 4}{x (x + 4)} = 0$

Step 1.7.6

Move $4$ to the left of $x$ .

$\frac{- 100 x + 1280 + x^{2} + 4 \cdot x}{x (x + 4)} = 0$

Step 1.7.7

Add $- 100 x$ and $4 x$ .

$\frac{- 96 x + 1280 + x^{2}}{x (x + 4)} = 0$

Step 1.7.8

Reorder terms.

$\frac{x^{2} - 96 x + 1280}{x (x + 4)} = 0$

Step 1.7.9

Factor $x^{2} - 96 x + 1280$ using the AC method.

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Step 1.7.9.1

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 $1280$ and whose sum is $- 96$ .

$- 80, - 16$

Step 1.7.9.2

Write the factored form using these integers.

$\frac{(x - 80) (x - 16)}{x (x + 4)} = 0$

Step 2

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

$x \approx 16$

Step 3