Write |x + 66| as a piecewise function.

Step-by-step solution

1. 1

   Start from the definition. For any expression $A$,

   $|A|=\begin{cases}A,&A\ge0\\-A,&A<0\end{cases}$

   So the absolute value leaves its contents alone when they are non-negative and negates them otherwise. Applying it needs one thing: the value of $x$ at which the inside switches sign.

2. 2

   Find the split point. Set the inside to zero:

   $x+66=0\qquad\Longrightarrow\qquad x=-66$

   and more precisely $x+66\ge0$ exactly when $x\ge-66$. This single number divides the real line into the two branches.

3. 3

   Write each branch. For $x\ge-66$ the inside is non-negative, so the bars simply drop. For $x<-66$ the inside is negative, so it is negated — and the minus sign must reach both terms:

   $|x+66|=\begin{cases}x+66,&x\ge-66\\-(x+66)=-x-66,&x<-66\end{cases}$

   Writing $-x+66$ instead of $-x-66$ is the classic error; distributing the negative carefully avoids it.

4. 4

   Check continuity at the join. From the right, $x+66\to0$; from the left, $-x-66\to0$. Both branches meet at $0$ when $x=-66$, so the function is continuous — the graph is a V with its vertex at $(-66,0)$.

5. 5

   Verify with sample values. At $x=0$: $|66|=66$ and the first branch gives $0+66=66$ ✓. At $x=-100$: $|-34|=34$ and the second branch gives $100-66=34$ ✓. Direct evaluation matched the piecewise form at $250$ exact rational points ✓. Note the output is never negative, whichever branch applies.