Question

Solve the equation and show all work. 4|x + 1/3| = 20
i'm kind of stuck hhh

Answer 1

Absolute value is a function that takes some number and outputs the size of that number, regardless of the sign. By definition, $|x|=x$ if $x\ge0$, and $|x|=-x$ if $x<0$.

For example,

• |2| = 2 because the input 2 is positive, so the input is unchanged;
• |-3| = -(-3) = 3 because the input -3 is negative, so the sign is switched.

What this means for this problem is that there are two cases to consider:

• If $x+\frac13\ge0$, then $\left|x+\frac13\right|=x+\frac13$, so that

$4\left(x+\dfrac13\right)=20\implies x+\dfrac13=5\implies\boxed{x=\dfrac{14}3}$

• On the other hand, if $x+\frac13<0$, then $\left|x+\frac13\right|=-\left(x+\frac13\right)$, so

$-4\left(x+\dfrac13\right)=20\implies x+\dfrac13=-5\implies\boxed{x=-\dfrac{16}3}$