Confused about equations with absolute values

[Nat3]

My calc book rewrites this equation:

$$|y|=e^c|x|$$

As this:

$$y=\pm e^cx$$

But that doesn't really make any sense to me. I know I should understand why we're allowed to do that, but I don't. Could someone please try to explain it to me?

I really appreciate your help, thanks!

 

[Nat3]

Is it because:

$$|y| = e^c|x| = |e^cx|$$

And there are four cases:

$$y = e^cx$$

$$y = -e^cx$$

$$-y = e^cx$$

$$-y = -e^cx$$

With the inner two and outer two being equivalent, respectively, resulting in:

$$y = e^cx$$

$$y = -e^cx$$

Which can be written as (?):

$$y = \pm e^cx$$

 

[The_Duck]

First consider the simpler equation

$$|y|=|x|$$

Suppose you know the value of x. What values of y would make the equation true?

 

[HallsofIvy]

Nat3, in LaTex, use { } to group. That is, use e^{cx} to get $e^{cx}$. e^cx gives $e^cx$.

 

[CellCoree]

I completely understand your confusion about equations with absolute values. It can be a tricky concept to grasp, but let me try to explain it to you.

First, let's define what absolute value means. Absolute value is the magnitude of a number, regardless of its sign. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.

Now, let's look at the equation |y|=e^c|x|. The absolute value bars tell us that whatever is inside them must be positive. So, we can rewrite this equation as y=e^c|x|, because the absolute value bars ensure that y will be positive.

Next, we can use the property of absolute value that states |ab|=|a||b|. This means that we can separate the absolute value of a product into the product of the absolute values. So, we can rewrite the equation as y=e^c|x| as y=e^c|x|.

Finally, we can use the fact that x can be either positive or negative. This means that we can write x as x=\pm|x|, where the plus or minus sign will depend on the sign of x. Substituting this into our equation, we get y=e^c|x|=\pm e^cx. This is why your calc book rewrote the equation as y=\pm e^cx.

I hope this explanation helps you understand why we can rewrite equations with absolute values in this way. It's important to remember the properties of absolute value and the fact that x can be positive or negative. Keep practicing and you'll get the hang of it!