In Exercises 61–76, solve each absolute value equation or indicat...

Question

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.

2|4 - (5/2)x| + 6 = 18

Accepted Answer

Welcome back. I'm so glad you're here. We have this equation three times the absolute value of seven minus one third X and absolute value plus eight equals 11. And we want to solve for X. Well in order to do that first, we want to get the absolute value all by itself on one side of the equal sign. So let's subtract eight from both sides. And on the left it's three times the absolute value of seven minus one third X equals. And on the right it is three. Now let's divide both sides by three. On the left. Those threes cancel out. So we just have the absolute value of 7 -1 3rd x. And on the right hand side, three divided by three is one. Great. Now we always want to pause when we have the absolute value all by itself on one side of the equal sign and ask ourselves is the other side of the equal sign Positive or is it greater than or equal to zero? And yes one is greater than or equal to zero. So we can proceed, absolute value always has to be equal to something that is greater than or equal to zero. Can't have a negative number on the other side. So now that we proceed, we want to think about what is absolute value. And we recall that absolute value is the distance from zero on a number line. So in this case seven minus one third X is one unit away from zero in the positive side and it's also one unit away from zero and the negative side. So to get rid of those absolute value brackets, we can set seven minus one third X equal to negative one. And we can set 7 -1 3rd x equal to positive one. Now we're going to sell for X for both of those equations. Let's subtract seven from both sides and we have negative one third X equals negative one minus seven is negative eight And will multiply both sides by -3 and negative three times negative one third cancels out to a one. So we've just got X on the left hand side and negative eight times negative three is positive 24. There's one of them. Let's get the other equation solved. Subtract seven from both sides again and we've got negative one third X equals negative six. Again we'll multiply both sides by -3. And on the left we've just got X. And on the right negative six times negative three is 18. So our two answers are X equals 24 or X equals 18. We look at our answer choices and that matches with answer choice. D Well done. We'll catch you on the next one

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In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|4 - (5/2)x| + 6 = 18