Eric Nicholas K's answer to Shakir's Secondary 4 A Maths Singapore question.
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And the third "DO NOT do this", which I forgot to list down.
Number 3 of "DO NOT do this" list
(m + 11) (m - 5) < 0
m + 11 < 0 or m - 5 < 0
m < -11 or m < 5
You must be more careful whenever you approach a quadratic or a non-linear inequality.
Number 3 of "DO NOT do this" list
(m + 11) (m - 5) < 0
m + 11 < 0 or m - 5 < 0
m < -11 or m < 5
You must be more careful whenever you approach a quadratic or a non-linear inequality.
Date Posted:
4 years ago
Actually for ② , it is not really incorrect. Just a minor adjustment to be made.
(-m-3)² < 64
(-m-3)² < 8²
This means that |-m-3|< 8
Therefore
-8 < -m-3 < 8
-5 < -m < 11
5 > m > -11
Rewrite as -11 < m < 5
(-m-3)² < 64
(-m-3)² < 8²
This means that |-m-3|< 8
Therefore
-8 < -m-3 < 8
-5 < -m < 11
5 > m > -11
Rewrite as -11 < m < 5
Also note that the inequality sign is wrong in the given answer. It cannot be ≤ as that would be smaller or equal to, suggesting that the curve does intersect , which contradicts the question
Though technically not fully wrong, in 100% of the cases I encountered, every student whom I have seen going on that path always makes the same mistake.
Which is odd , as they have encountered similar examples whereby the sign is an equal sign :
(-m-3)² = 64
-m - 3 = 8 or -m - 3 = -8
There will be 2 possible values and for the inequality a minor adjustment is made:
for (-m-3)² > 64
-m - 3 > 8 or -m - 3 < - 8
Likewise for (-m - 3)² < 64 as shown in the previous comment
(-m-3)² = 64
-m - 3 = 8 or -m - 3 = -8
There will be 2 possible values and for the inequality a minor adjustment is made:
for (-m-3)² > 64
-m - 3 > 8 or -m - 3 < - 8
Likewise for (-m - 3)² < 64 as shown in the previous comment
Those students I have observed always do this.
Some of them know that (-m - 3)^2 = 64 means (-m - 3)^2 = +-8. Even this one, the rest forget to put +-.
Surprisingly all of them, when they see (-m - 3)^2 < 64, either put -m - 3 < 8 or they put “-m - 3 < 8 or -m - 3 < -8”.
Basically they conveniently replace “=“ by “<“ without making other adjustments.
Some of them know that (-m - 3)^2 = 64 means (-m - 3)^2 = +-8. Even this one, the rest forget to put +-.
Surprisingly all of them, when they see (-m - 3)^2 < 64, either put -m - 3 < 8 or they put “-m - 3 < 8 or -m - 3 < -8”.
Basically they conveniently replace “=“ by “<“ without making other adjustments.
Yes, surprising indeed, when it isn't difficult for them to rationalise why.
When (-m - 3)² < 8²
If -m - 3 is is negative, it has to be bigger than -8 in order for its square to be smaller than 64.
Eg. Let's say -m - 3 = -7,
then (-m - 3)² = (-7)² = 49 < 64
If -m - 3 is is positive, it has to be smaller than 8 in order for its square to be smaller than 64.
Eg. Let's say -m - 3 = 6
then (-m - 3)² = 6² = 36 < 64
The concept of absolute value (modulus) has to be grasped correctly for this
When (-m - 3)² < 8²
If -m - 3 is is negative, it has to be bigger than -8 in order for its square to be smaller than 64.
Eg. Let's say -m - 3 = -7,
then (-m - 3)² = (-7)² = 49 < 64
If -m - 3 is is positive, it has to be smaller than 8 in order for its square to be smaller than 64.
Eg. Let's say -m - 3 = 6
then (-m - 3)² = 6² = 36 < 64
The concept of absolute value (modulus) has to be grasped correctly for this
For these groups of people, the thought processes are not so automatic.
They can probably identify that for negative numbers, the square of a more negative number is greater than the square of a less negative number, but when it comes to this question, these thoughts do not come instantly and they may not even realise it.
They can probably identify that for negative numbers, the square of a more negative number is greater than the square of a less negative number, but when it comes to this question, these thoughts do not come instantly and they may not even realise it.
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