In Exercises 101–106, solve each equation.

|√x - 5| = 2

Question

In Exercises 101–106, solve each equation.

|√x - 5| = 2

Accepted Answer

Hello today I'm going to be solving the absolute value equation. So the equation given to us is the absolute value of two times the square root of X minus 11 is equal to five. Now in order to solve for X, I need to go ahead and open up this absolute value quantity. And in order to do that I'm going to set the two equations. One word, the absolute value quantity is equal to positive five and one where the absolute value quantity is equal to negative five. So I'm left with two times the square root of X minus 11 is equal to five and two times the square root of X minus is equal to negative five. And we need to go ahead and sulfur X for both of these equations. So let's start with the left hand side On the left hand side. I'm gonna go ahead and add 11 to both sides of this equation in order to isolate X. And doing so is going to leave me with two times the square root of X. Is equal to 16. Now I'm gonna divide both sides of this equation by two. And that's gonna leave me with, the square root of X is equal to eight. And in order to get X out of the square root, I'm gonna square both sides of this equation. And that's going to leave me with X is equal to eight squared which is 64. And that's my first value for X. Not going to solve the right hand side. Once again I'm gonna go ahead and add 11 to both sides of this quantity. This equation Doing so is gonna leave me with two times the square root of X is equal to six. And if I divide both sides of this equation by two, I am left with the square root of X is equal to three. And once again I'm going to square both sides of this equation to get X out of the square root and I am left with X is equal to three squared which is nine and that is my other value of X that I solved for now. I need to verify that both both values of this X work with our original equation. So when X is equal to 64 I get the absolute value of two times the square root of 64 -11 is equal to five. Now the square root of 64 evaluates to eight. So what I am left with is two times eight minus is equal to five. While two times eight leaves me with 16 and minus 11 gives me the absolute value of five is equal to five. Now again, the absolute value allows you to express any number as a positive quantity. So the absolute value of five is equal to five and we are left with five is equal to five which is a true statement. So X is equal to 64 is one of our solutions. Now we need to check to see if X is equal to nine works with our original equation. So when X is equal to nine, we have the absolute value of two times the square root of nine minus 11 is equal to five. The square root of nine evaluates the three. So we are left with two times three minus 11 is equal to five. Now two times three gives me six and six minus gives me negative five. So I have the absolute value of negative five is equal to five and once again the absolute value, let's me represent any number as a positive number. So the absolute value of negative five gives me five and I am left with five is equal to five which is a true statement and that means that X is equal to nine is our other solution for X. So the two solutions for X that we have solved for our X is equal to 64 and X is equal to nine. And that means that the answer to this problem is going to be C. So I hope this video helps you in understanding how to solve an absolute value equation. And I'll go ahead and see you all in the next video

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