If $ a + b = 4$, what is the value of $|4- a|+|4-b|$ ? ( $-4$ ? , $0$ ? , $4$ ? , $8$ ? , or impossible to determine? )

Question

Source : CLEP COLLEGE ALGEBRA TEST http://www.nelnetsolutions.com/pdf/clep_collegealgebra_q_print.pdf

If $ a + b = 4$, what is the value of $|4- a| +|4-b|$ ? ( Possible answers : $-4$ ? , $0$ ? , $4$ ? , $8$ ? , or impossible to determine? )

I tried this

(1) If $ a + b = 4$ then

• $ a + b = | a + b |$

• $a = 4 - b$

(2) So $|4- a| +|4-b| = | 4 - (4 - b)| + |4 - b| = |b| + |a|$

(3) By triangle inequality reversed : $|b| + |a| \geq |a+b|$ ( $= a+b$ , by (1) ).

(4) So $|4- a| +|4-b| \geq 4$.

Answer : impossible to determine.

Answer 1

If $a+b = 4$, then $b = 4-a$ and $a = 4-b$, so $$|4-a|+|4-b| = |a|+|b|.$$ Now, if both $a$ and $b$ are nonnegative, then $$|a|+|b| = a + b = 4.$$ However, it can also be the case that one of them is negative, WLOG, let it be $b$. Then, $a>4$ and $$|a|+|b| = a -b = a - (a-4) = 2a - 4 > 8 - 4 = 4.$$ We conclude that $|4-a|+|4-b|$ can obtain any value greater or equal than $4$, and thus is impossible to determine from the given data.

Answer 2

Impossible to determine. Just try a couple different examples. $(a,b)=(4,0), (-1,5)$ should give you different results

Answer 3

Since we have $b = 4 - a$, we can draw a graph of $|4-a| + |4-(4-a)|$ to see what happens.

If we enter the formula on Wolfram Alpha, we get the following graph:

Since $|4 - (4 - a)| = |a|$, the formula is equal to $|4 - a| + |a|$, which can be interpreted as distance to 4 + distance to 0.