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# In mathematics, an absolute value is a value that is considered to be independent of the location or direction of a given vector or number in space. The absolute value of a real number is its numerical value. For example, the absolute value of the number 4 is 4, regardless of the direction of the number or the location of the origin.

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we're being asked to solve to given equation. So our equation is the absolute value of X plus four. Minus seven is equal to three now. No, this the left hand side of our equation. We have the absolute value of X Plus four, but it's not by itself because it's getting subtracted by seven. So the first thing we have to do is isolate the absolute value. Simple. So the question is, how do we move the minus seven? Well, just like when you're solving regular equations, we're going to do the inverse operation, which means we're going to add seven. Tow both sides of our equation because negative seven plus seven is zero. So these terms cancel up, so we're gonna bring down the absolute value of X plus four, and then it's equal to while three plus seven is 10. So now we have isolated are absolute value, which is exactly what we need to dio. Now we have the absolute value of X plus four, and it's equal to a positive number. Therefore, we will have two solutions. So let's set up our two equations. So remember to get our first equation. We're gonna take what's inside the absolute value simple, which is X Plus four and said it equal to the number on the outside, which is positive. 10. Now our second equation again, We're going to still stay with what's inside our absolute value symbol, which is still X plus four. But remember, it's gonna be equal to we're going to do the opposite of the number on the outside. So negative 10. And now we've set up our two equations. Now we just need to go ahead and solve. So let's start with the equation on the right. Well, to get X by itself, we're going to subtract four from both sides and 10 minus four is six so well found that one of our solutions is that X is equal to six. Now let's solve the equation on the right. Well, the solve for X. We're also going to subject for on both sides. However, this time we have negative 10 minus four, which is negative. 14. So are our solution is negative 14 and now we've solved our equation. So are two solutions are six and negative. 14. Like always, you could go back and plug them both into your original problem, and you'll find that it will make both equations true. So the key thing to this example is when you're given equations and there's values or terms on the outside, the absolute value simple, you have to move them to the other side first, before you can set up your two equations.

Syracuse University