Hel! 1. Explain why the equations below do not have the same solution set. (5 points)

|x| = a and |x| = -a

(Hint: Remember that the
definition of absolute value is related to a number's distance from zero.)

Let's say that 'a' is some positive number. Let's pick a = 5.

That means |x| = a turns into |x| = 5

The equation |x| = 5 solves to x = -5 or x = 5.

This is because the values -5 and 5 on the number line are exactly five units away from zero.

In short, a = 5 leads to |x| = a having the two solutions x = -5 or x = 5.

Keeping the same value of 'a', the equation |x| = -a turns into |x| = -5

What number on the number line is exactly negative 5 units away from zero? The answer is "no such number exists". Distance is never negative.

Therefore, |x| = -5 has no solutions.

So if a = 5, then the first equation |x| = a has two solutions, while |x| = -a has no solutions.

If we made 'a' to be some negative number, then things would flip around. In this case, it would mean |x| = a has no solutions while |x| = -a has two solutions.

The only time both equations would have a solution is when a = 0

We can see that |x| = a becomes |x| = 0, and |x| = -a becomes |x| = -0 or just |x| = 0 which is the exact same thing the first equation turned into. We're dealing with the same equation at this point.

If your teacher states "The value of 'a' is nonzero", then the two equations do not have the same solution set. If a = 0, then it leads to x = 0 as the only solution for both equations.