Question

For what values of a are the following expressions true?
1. |a-5|=5-a
2.|a|=-a

Answer 1

1. a = 0 and 5

2. no solutions

Explanation:

For absolute value, it's the distance from that number on the number line to 0, so it has to be positive. We define absolute value as when |a| = b, a = b and a = -b.

1. |a - 5| = 5 - a

a - 5 = 5 - a

2a = 10

a = 5

AND

a - 5 = -(5 - a)

a - 5 = -5 + a

-5 = -5

a = 0 (a could work as 0 in this scenario)

Thus our solutions are a = 0 and 5.

2. |a| = -1

Now we have to catch ourselves here. Since absolute value is always positive, nothing works. There is no negative distance to 0 (at least in real numbers). Hence, we can say that there are no solutions.