Question

Please review the following attachment. Thanks!

Answer 1

Answer:

C: I and III only.

Step-by-step explanation:

We have two numbers x and y such that:

$-y$

And we want to determine which of the following conditions must be true.

First, let's examine the third condition. We have:

$y>0$

To see if this is true, let's try a negative value for y. So, let's use -7. This will give:

$-(-7)$

Simplify:

$7$

This is saying we need a number less than -7 and greater than 7.

This is impossible. So, y must be greater than 0.

Therefore, Condition III must be true.

Next, let's see this compound inequality visually.

Picture the following number-line:

<----------(-y)----------0----------y---------->

Whatever y is, -y is just y on the negative side.

Now, since our condition is that -y<x<y, this means that our x can be anywhere between -y and y. Namely:

<----------(-y)----------0----------y---------->

To determine our correct condition, let's picture x anywhere on the bolded lines. I'm just going to put x here...

<----------(-y)-------(x)--0----------y---------->

Now, remember the alternative definition for absolute value. Namely, the absolute value of x is also the distance from 0 to x. And this distance is always positive.

Since our x is between -y and y, our distance from 0 to x will always be less than the distance from 0 to either y.

Therefore, Condition I is also true.

Let's try an example. Let's let y = 9. So, -y=-9. And let's let x be between 9 and -9, say, -2. So:

<---(-9)-----(-2)--0--------(9)------>

We can see that:

$|-2|=2\stackrel{\checkmark}{$

Also, this example counters Condition II, as our x can indeed be negative if we desire it.

Algebraically, if we have a negative x such that:  

$-y$

We can divide everything by -1 to obtain:

$y>x>-y$

We can flip this to get:

$-y$

Which is our original inequality. So, Condition II does not need to be true.

So, the two conditions that must be true is I and III.

So, our answer is C.