# Confused about equations with absolute values

• Nat3
In summary, the conversation discusses the rewriting of an equation from |y|=e^c|x| to y=\pm e^cx and the confusion the speaker has regarding the reasoning behind it. The idea is that |y| and e^c|x| are equivalent, leading to four possible cases which can be simplified to two solutions, resulting in y=\pm e^cx. The conversation also mentions the use of grouping in LaTex to properly format the equation.
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!

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 (?):

$$y = \pm e^cx$$

First consider the simpler equation

$$|y|=|x|$$

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

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

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!

## 1) What is an absolute value in an equation?

An absolute value in an equation represents the distance between a number and 0 on a number line. It always results in a positive value.

## 2) Why do equations with absolute values have two solutions?

Equations with absolute values can have two solutions because the absolute value could be positive or negative, depending on the value of the variable. This results in two possible solutions.

## 3) How do I solve an equation with absolute values?

To solve an equation with absolute values, you need to isolate the absolute value, and then split the equation into two separate equations: one with the positive value and one with the negative value. Finally, solve for the variable in each equation to find the solutions.

## 4) Can an equation with absolute values have no solution?

Yes, an equation with absolute values can have no solution. This occurs when the absolute value expression results in a negative value, which is not possible.

## 5) How do I check my solutions for an equation with absolute values?

To check your solutions for an equation with absolute values, plug in the values for the variable in the original equation and see if both sides of the equation are equal. If they are, then the solution is correct.

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