Two different points on a number line are both seven units from the point with coordinate negative six. Which of the following equations has a solution that gives the coordinates of both points? A) The modulus or absolute value of 𝑥 minus six is equal to seven, B) the absolute value of 𝑥 plus six is equal to seven, C) the absolute value of 𝑥 plus seven is equal to six, or D) the absolute value of 𝑥 minus seven is equal to six.
So to help us understand this question, what I’ve drawn is our number line. And I’ve marked on negative six. And we’re told that we have two points on the line that are seven units from negative six. So the first point would be seven units to the left or subtract seven from negative six. And that takes us to negative 13. And the other point is gonna be seven units to the right or we add seven units on from negative six. So this will take us to one.
Okay, great, so we found our two points. But what we’re looking for is an equation that will give us the solutions that will then give us these two points. Well, to help us show our equation, what we’re gonna be using is something here that I’ve shown around the 𝑥. And that are two vertical lines. And these two vertical lines tell us that we’re looking at the modulus or absolute value of 𝑥 in this case.
And what the modulus or absolute value are are distances from a point. And that’s the way you can think about it because we’re not interested here in being positive or negative. We’re just saying that there’s a distance from a point. And therefore, if we use this thinking to help us write our equation, we can say that the modulus or absolute value of 𝑥 plus six must be equal to seven. And this is where 𝑥 represents our two points. And that’s because if we consider this equation, then we’ve got 𝑥, which is our point. And then we’re adding on a six, because we’re just thinking here in terms of units. We’re not thinking about the actual values.
So if we have our point and then we add on six, then the modulus of this or the absolute value is gonna be the distance that there is between our point and our coordinate, which is seven, because we’re told that in the question. So it’s quite some tricky possibly to understand this kind of concept. So a good way to actually show it is to have a look at how the equation will work.
Well, if we have our equation, which is the same equation as equation B, then we’re gonna have the absolute value of 𝑥 plus six is equal to seven. And when we’re solving one of these types of equations, what we need to do is consider both possible answers. And that is positive and negative seven, because once we remove the absolute value signs, then we have to then consider positive and negative answers.
So the two equations that we need to co- consider are 𝑥 plus six is equal to seven or 𝑥 plus six is equal to negative seven. So if we look at the solution to the left-hand equation, we’re gonna have 𝑥 equals one. And that’s because if we subtract six from both sides of the equation, we get 𝑥 equals one. And then the equation on the right-hand side, we’ve got 𝑥 plus six equals negative seven.
If we subtract six from each side of this equation, we’re gonna get 𝑥 equals negative 13. So if we take a look back at our number line, we can see that the two solutions are the solutions that we expected. So 𝑥 equals one and 𝑥 equals negative 13, as these are both seven units away from the point negative six. So it confirms the correct answer is B. The modulus or absolute value of 𝑥 plus six equals seven.
And what we could do is to use the same method to check the other answers to make sure that we were correct. And to do the checks, you wouldn’t have to do the full calculations because you could straight away see that if you’ve got the absolute value of 𝑥 minus six is equal to seven, then one of the solutions is gonna be 𝑥 minus six is equal to seven. Well, if we add six to each side of that equation, we get 𝑥 equals 13, which is not one of the values we’d expect on the number line.
Similarly, if we look at C, we’ve got the absolute value of 𝑥 plus seven is equal to six. So that means that one of our solutions would be 𝑥 plus six equals seven. So therefore, take seven away from each side of the equation, we get 𝑥 equals negative one. Again, isn’t one of the numbers that we’re looking for on our number line.
And then, finally, for D, we had the absolute value of 𝑥 minus seven equals six. So one of our solutions here would be 𝑥 minus seven is equal to six. So therefore, add seven to each side of the equation, we get 𝑥 is equal to 13, again not one of the numbers on our number line that we’d be looking for. So therefore, we’ve confirmed that the correct answer is B. The absolute value of 𝑥 plus six is equal to seven.