Modulus Question

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taylorswift

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Hi I’ve completed this functions question. I’m pretty confident on part a and b, but I’m not sure if part c is right. Please could someone check my workings and verify if I have used set notation correctly? TIA
 

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taylorswift said:
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Hi I’ve completed this functions question. I’m pretty confident on part a and b, but I’m not sure if part c is right. Please could someone check my workings and verify if I have used set notation correctly? TIA
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So, you’re saying that the solution is the set of all numbers that are greater than 2 and less than 1.25? How many numbers are there that satisfy this condition? Think about it, don’t just blindly write out your results, try to test them ?. Also, while your logic behind solving this problem generally makes sense, you should put more effort in explaining (proving) why this has to hold. Personally, I would approach this problem a little differently, if the two lines intersect, then there must exist a point (x,y) that satisfies both equations. Solve y1=y2 for x, where y1 and y2 are the respective line equations, and conclude for which “a” there exists a solution.
 
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Clearly the two graphs intersect if the line y=ax passes 'above' (not below) the point P.

If y=ax is above the point (-4,-5) then -4a≥-5 i.e. [MATH]\boxed{\hspace1ex a≤1.25 \hspace1ex}[/MATH]
Now we only have to look at lines (with positive gradients) with a>1.25
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Clearly there is not another intersection before the gradient becomes 2 (matching the gradient of the right half of the graph), so also [MATH]\boxed{ \hspace1ex a>2 \hspace1ex}[/MATH] ensures an intersection.
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So you can now write the full solution set in set notation.

Btw we've been here before: https://www.freemathhelp.com/forum/threads/functions.129669/
 
Thanks you’ve been more helpful. However my original question was about set notation. I have two answers and I’m not sure if either is correct.
 

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The 1st solution says that a>2 AND a<= 5/4. What number, numbers, if any satisfy these two condition.

You do realize that if a<b<c, then a<c. Then your solution says that 2<5/4. My question at this point is always how much less is 2 than 5/4?

Your 2nd solution states that a>2 OR a<=5/4.

Which of these two, if any, satisfies your problem?
 
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The solution set consists of 2 intervals, so it needs to be stated with the two inequalities I have 'boxed' in my post #4. [MATH]a \in \mathbb{R}[/MATH] can be: a≤1.25 OR a>2
The problem with your first attempt was that since there is only one 'a', you are requiring the same number to be both greater than 2, and simultaneously ≤ 1.25.
 
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