I don't understand where 4x-3< -5 comes from... I tried using the graphical method, but this gives me an answer of x = 2 and x = 1/2. My first answer is correct but the second answer should be -1/2... Can someone please explain this to me... thanks in advance
I don't understand where 4x-3< -5 comes from... I tried using the graphical method, but this gives me an answer of x = 2 and x = 1/2. My first answer is correct but the second answer should be -1/2... Can someone please explain this to me... thanks in advance
Comes from the fact that ∣4x−3∣>5 means 4x−3>5 or −(4x−3)>5 so the second one mult both sides by -1 and get 4x−3<−5
If the distance must be greater than five, how can it also be less than -5 simultaneously...?
Never said the distance was less than -5. I said that any number smaller than -5 has a distance greater than 5 from the origin. The displacement is less than -5, but the distance is still > 5.
Essentially, if z = -5.1 then the distance from z to 0 is the length of the line segment connecting 0 and z. Length is a positive quantity. Any z > 5 will have distance > 5 from the origin. Any z < -5 will also have distance greater than 5 from the origin.
If the distance must be greater than five, how can it also be less than -5 simultaneously...?
If the above still doesn't click for you, then maybe thinking about it in terms of "steps" will help. 1 step = 1 integer. So |z| >5 means that you need to take more than 5 steps to reach z (starting from the origin). If z>5 then you need to take at least 5 steps to the right to reach z. If z < -5, you still need to take at least 5 steps, just that it's to the left this time. But you're still taking at least 5 steps, so any z < -5 will have |z| > 5.
I don't understand where 4x-3< -5 comes from... I tried using the graphical method, but this gives me an answer of x = 2 and x = 1/2. My first answer is correct but the second answer should be -1/2... Can someone please explain this to me... thanks in advance
Some student prefer to think about squaring both sides:
(4x - 3)^2 > 25 then solve the quadratic inequality.
If the above still doesn't click for you, then maybe thinking about it in terms of "steps" will help. 1 step = 1 integer. So |z| >5 means that you need to take more than 5 steps to reach z (starting from the origin). If z>5 then you need to take at least 5 steps to the right to reach z. If z < -5, you still need to take at least 5 steps, just that it's to the left this time. But you're still taking at least 5 steps, so any z < -5 will have |z| > 5.
This makes more sense to me than the previous explanation... but I still don't understand why the 5 must be made negative too when the modulus sign is only around 4x-3??
I don't understand where 4x-3< -5 comes from... I tried using the graphical method, but this gives me an answer of x = 2 and x = 1/2. My first answer is correct but the second answer should be -1/2... Can someone please explain this to me... thanks in advance
∣4x−3∣=∣−(4x−3)∣=4x−3
If we gave 4x-3 a value, we would still have two moduli equal to it, so we have to consider both as they both satisfy the equation.
Remember that modulus functions are piecewise and we always have to consider both branches - f(x) and −[f(x)]
Some student prefer to think about squaring both sides:
(4x - 3)^2 > 25 then solve the quadratic inequality.
Does that help?
Oh yes, I like that method but I want to make sure I understand the other method too since I can't always use the squaring method. Thank you for your input though
This makes more sense to me than the previous explanation... but I still don't understand why the 5 must be made negative too when the modulus sign is only around 4x-3??
I'm not really sure what you're saying. We have an inequality |z| > 5. We're trying to find the all the numbers z such that this is true. We've established that it means that z must be a number bigger than 5 or a number smaller than -5.
So z > 5 or z < -5. Now we replace z = 4x-3 and solve the resulting inequalities.
I'm not sure why you seem to be associating the modulus sign with having to shove a minus sign there for no reason. We're not putting minus signs anywhere formulaically, we're reasoning geometrically why |z| > 5 means that z > 5 or z < -5.
Oh yes, I like that method but I want to make sure I understand the other method too since I can't always use the squaring method. Thank you for your input though
OK - no worries.
Just remember the modules signs mean the 'positive value of' so we are solving
+ (4x - 3) > 5
and - (4x - 3) > 5 which is the same as - 4x + 3 > 5 3 - 5 > 4x
[I prefer this to changing direction of the inequality and lots of students don't understand this]
I don't understand where 4x-3< -5 comes from... I tried using the graphical method, but this gives me an answer of x = 2 and x = 1/2. My first answer is correct but the second answer should be -1/2... Can someone please explain this to me... thanks in advance
Try thinking about it this way. The modulus gives us the magnitude of something, the 'distance' it is from 0. So the first case is when 4x - 3 > 5, as everything above 5 is at a greater distance from zero.
The case you are confused about, 4x - 3 < -5, is because everything below -5 is getting 'further away' from 0. E.g -6 has modulus 6 and so on. Picture the number line if it seems confusing.
In case the above makes no sense at all, the alternative is to just remember the definition of the modulus of a number. The modulus is the number itself (when the number is 0 or more) and the negative of the number when it is less than 0.
META: Unless the OP asks for other people to chime in, I'd suggest other people stop adding alternative viewpoints . I think this is becoming a too many cooks situation right now TBH.
META: Unless the OP asks for other people to chime in, I'd suggest other people stop adding alternative viewpoints . I think this is becoming a too many cooks situation right now TBH.
Why? The OP was not understanding the posts so needed better/different explanations.
Why? The OP was not understanding the posts so needed better/different explanations.Well, as you ask, the OP seemed to be doing quite well with your suggestions, I don't think there's enough evidence that they're still so stuck as to think subsequent suggestions from different posters would be terribly helpful. (Note that I haven't made any suggestions myself for the same reason).
I'll note that one of the proposed new rules for this forum states (albeit only as an advisory guideline):
7) Don't "barge in" while another helper is offering hints. If you notice that a student and helper are replying to each other back-and-forth then it's best to allow this one-on-one to continue without interrupting, unless you feel that the helper has made a mistake.
Well, as you ask, the OP seemed to be doing quite well with your suggestions, I don't think there's enough evidence that they're still so stuck as to think subsequent suggestions from different posters would be terribly helpful. (Note that I haven't made any suggestions myself for the same reason).
I'll note that one of the proposed new rules for this forum states (albeit only as an advisory guideline):
So who do you think 'barged in' here? I'm not clear who your post is aimed at.