Question

which expression(s) are equivalent to |9 - x|?
a) |x| - |9| if x > 9
b) |x - 9|
c) |9| - |x| if x > 9
d) |9| + |x| if x ≤ 0
e) |x| - |9| if 9 > x

Explanation:

Step1: Analyze Option B

Recall $|a-b|=|b-a|$.
$|9-x|=|-(x-9)|=|x-9|$, so B is equivalent.

Step2: Analyze Option A ($x>9$)

If $x>9$, $9-x<0$, so $|9-x|=x-9$.
Also $|x|-|9|=x-9$. Thus A is equivalent.

Step3: Analyze Option C ($x>9$)

$|9|-|x|=9-x$, but $|9-x|=x-9$.
$9-x
eq x-9$ for $x>9$, so C is not equivalent.

Step4: Analyze Option D ($x\leq0$)

If $x\leq0$, $9-x>0$, so $|9-x|=9-x$.
$|9|+|x|=9+(-x)=9-x$. Thus D is equivalent.

Step5: Analyze Option E ($9>x$)

If $9>x$, $9-x>0$, so $|9-x|=9-x$.
$|x|-|9|=x-9$. $9-x
eq x-9$ for $x<9$, so E is not equivalent.

Answer:

B) $|x - 9|$, A) $|x| - |9|$ if $x > 9$, D) $|9| + |x|$ if $x \leq 0$