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

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Answer 1 · 2017-08-08T20:58:27

See a solution process below:

We can write this proportion as:

$\frac{x}{x + 1} = \frac{4}{x + 4}$ $x/(x + 1) = 4/(x + 4)$

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

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$x (x + 4) = 4 (x + 1)$ $x(x + 4) = 4(x + 1)$

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

We can now put this in standard form:

$x^{2} + 4 x - 4 x - 4 = 4 x - 4 x + 4 - 4$ $x^2 + 4x - color(red)(4x) - color(blue)(4) = 4x - color(red)(4x) + 4 - color(blue)(4)$

$x^{2} + 0 - 4 = 0 + 0$ $x^2 + 0 - color(blue)(4) = 0 + 0$

$x^{2} - 4 = 0$ $x^2 - color(blue)(4) = 0$

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

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

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

Solution 1

$x + 2 = 0$ $x + 2 = 0$

$x + 2 - 2 = 0 - 2$ $x + 2 - color(red)(2) = 0 - color(red)(2)$

$x + 0 = - 2$ $x + 0 = -2$

$x = - 2$ $x = -2$

Solution 2

$x - 2 = 0$ $x - 2 = 0$

$x - 2 + 2 = 0 + 2$ $x - 2 + color(red)(2) = 0 + color(red)(2)$

$x - 0 = 2$ $x - 0 = 2$

$x = 2$ $x = 2$

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