Solve the proportion x over x plus 1 equals 4 over x plus 4. What is the value(s) of x?

Aug 8, 2017

See a solution process below:

Explanation:

We can write this proportion as:

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

Next, we can do cross product or cross multiply the equation:

$x \left(x + 4\right) = 4 \left(x + 1\right)$

${x}^{2} + 4 x = 4 x + 4$

We can now put this in standard form:

${x}^{2} + 4 x - \textcolor{red}{4 x} - \textcolor{b l u e}{4} = 4 x - \textcolor{red}{4 x} + 4 - \textcolor{b l u e}{4}$

${x}^{2} + 0 - \textcolor{b l u e}{4} = 0 + 0$

${x}^{2} - \textcolor{b l u e}{4} = 0$

Then, the left side of the equation is a difference of squares so we can factor it as:

$\left(x + 2\right) \left(x - 2\right) = 0$

Now, to find the values of $x$ we solve each term on the left side for $0$:

Solution 1

$x + 2 = 0$

$x + 2 - \textcolor{red}{2} = 0 - \textcolor{red}{2}$

$x + 0 = - 2$

$x = - 2$

Solution 2

$x - 2 = 0$

$x - 2 + \textcolor{red}{2} = 0 + \textcolor{red}{2}$

$x - 0 = 2$

$x = 2$

The Solutions Are: $x = - 2$ and $x = 2$