WEBVTT

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Hi, I'm Fiona.

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And here we're going to look at TAMUA
2018 paper one question 10.

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We're told X&
Y satisfy the modulus of 2 -,

5168c864-6cb4-4d4d-8c28-d9f2c6ec7960-1
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X less than or equal to six and the
modulus of y + 2 less than or equal to

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four.

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And we're asked what's the greatest
possible value of the modulus of X * y.

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Now for this question,
it's important to be very familiar and

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confident with modulus functions,
so I'm going to start by drawing the

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graph of a basic modulus function.

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So here's a graph of the function Y
equals the modulus of X,

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and when I want to consider what it means
for the modulus of X to be less than or

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equal to some value,
let's call that value a.

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Here on our graph.

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That's the same as saying for what values
of X does the function lie below or on

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the line y = a.

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And here I can see that because the
gradient of my function for positive

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values of X is 1 and for negative values
of X is -1,

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then the value of X at this point in my
function is minus A and the value of X at

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this point in my function is a positive A.

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So now hopefully you can see that in
order for the modulus of X to be less

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than or equal to some value A,
that's equivalent to saying that X is

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between minus A&A.

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And I'm going to use this result on my
inequalities in X&

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Y to really get into this question.

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So in order for the modulus of 2 -,
X to believe be less than or equal to six,

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well, that's equivalent to 2 -,
X being between -6 and +6.

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And now I've got an inequality in X that
doesn't involve any modulus signs,

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and so I'm going to take two from each
component of this inequality.

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I'm going to multiply all across by -1
and the values that X can take are

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between -4 and 8.

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Now I'm going to apply the same result to
my inequality in Y,

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and I can see that my possible values for
Y are between -6 and 2.

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So now just wrapping this up.

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In order to find the greatest possible
value of the modulus of X&

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YI just need to choose the right values
for X and the right values for Y.

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So I'm going to choose a value of eight
for X and a value of -6 for Y.

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So the modulus of 8 * -,
6 equals the modulus of -48,

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which will give me a +48.

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And now looking at the options I have to
choose from for my final answer,

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I can see that option A is a value of 48.

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And so that's the answer to this question.

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Now let's take some time to reflect on
this question.

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And I want to focus on this result that
the modulus of X less than or equal to a

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is equivalent to X being between minus
A&

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A even if we just take the X axis or a
kind of 1 dimensional number line.

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Let's say 0 is here.

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Then I have one 2-3 in the positive
direction -1 -, 2 -,

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3 in the negative direction.

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Then the modulus of X being less than or
equal to some value,

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I'm going to choose a value here,
let's say 2.

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That's just telling me that the distance
that X is away from zero can be at most 2.

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And if that's true in the positive
direction,

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then it's also true in the negative
direction.

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And so looking at it this way,
or considering X,

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considering the modulus as giving a
distance,

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we can also then see this relation that
this statement here is equivalent to X

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being between -2 and 2.