need help with non linear inequalities

[abel muroi]

I was given the problem x² < 7x - 10 and i was told to solve for x.

and here is the work I did.. i will write out the steps that I took so that someone can give me some feedback



x² < 7x - 10
-7x + 10 + x² < 0 ((i subtracted 7x and added 10 to both sides))
(x - 5) (x - 2) < 0 (( i tried to find two numbers that multiplied together to equal 10 and added up to -7x))

x > 2 or x < 5 ((this is my solution))



i tested out all the numbers between 2 and 5 by substituting those numbers for x and they all were less than 0.

did i make a mistake??

 

[Steven G]

I was given the problem x² < 7x - 10 and i was told to solve for x.

and here is the work I did.. i will write out the steps that I took so that someone can give me some feedback



x² < 7x - 10
-7x + 10 + x² < 0 ((i subtracted 7x and added 10 to both sides))
(x - 5) (x - 2) < 0 (( i tried to find two numbers that multiplied together to equal 10 and added up to -7x))

x > 2 or x < 5 ((this is my solution))



i tested out all the numbers between 2 and 5 by substituting those numbers for x and they all were less than 0.

did i make a mistake??

If x > 2 or x < 5 then which numbers are not included?

I claim x > 2 or x < 5 is equivalent to -infinity to infinity. Do you agree/disagree? Why?


Why did you only test values between 2 and 5?
How did you test ALL numbers between 2 and 5?
Did it take long??
Can you give me 4 numbers between 2 and 5?

What is the difference between x > 2 or x < 5 vs x > 2 and x < 5?


Please answer those questions.

 

[abel muroi]

If x > 2 or x < 5 then which numbers are not included?

I claim x > 2 or x < 5 is equivalent to -infinity to infinity. Do you agree/disagree? Why?


Why did you only test values between 2 and 5?
How did you test ALL numbers between 2 and 5?
Did it take long??
Can you give me 4 numbers between 2 and 5?

What is the difference between x > 2 or x < 5 vs x > 2 and x < 5?


Please answer those questions.

I claim x > 2 or x < 5 is equivalent to -infinity to infinity. Do you agree/disagree? Why?

i would disagree since those numbers are within a range i guess.


Why did you only test values between 2 and 5?

I didn't just test the numbers between 2 and 5, i also tested 6 and 1 and the result was bigger than 0. and I knew all the numbers that were smaller than 1 and bigger than 6 were going to result more than 0. So i didn't include those numbers.

How did you test ALL numbers between 2 and 5?

I tested the numbers by substituting those numbers for X. if the result was less than 0 then it was part of the solution

What is the difference between x > 2 or x < 5 vs x > 2 and x < 5?

they look the same to me

Can you give me 4 numbers between 2 and 5?

2,3,4,5, but i thought i shouldn't include the 2 and 5 since the equation has a < symbol

 

[stapel]

I was given the problem x² < 7x - 10 and i was told to solve for x.

x² < 7x - 10
-7x + 10 + x² < 0 ((i subtracted 7x and added 10 to both sides))
(x - 5) (x - 2) < 0 (( i tried to find two numbers that multiplied together to equal 10 and added up to -7x))

You're fine to this point. Now you've got a product of two factors, which evaluates to a negative number. What must be true for a product of two values to be negative? The signs on the two values in the product must be opposite; one must be positive and the other must be negative. This gives you TWO cases:

. . . . .1) x - 5 < 0 and (not "or") x - 2 > 0

. . . . .2) x - 5 > 0 and (not "or") x - 2 < 0

(List these two cases in your homework solution!)

Now consider each of the two cases.

1) If x - 5 < 0, then x < 5. If x - 2 > 0, then x > 2. These two constraints together mean that 2 < x < 5.

2) If x - 5 > 0, then x > 5. If x - 2 < 0, then x < 2. But x cannot be BOTH greater than 5 and ALSO less than 2. So there is no x-value which works for these conditions.

Thus, only the first case "works".

Note that this method will work for messy numbers and for products of more than two factors, since you'll be able to split things into cases and consider each case logically.

 

[Steven G]

I claim x > 2 or x < 5 is equivalent to -infinity to infinity. Do you agree/disagree? Why?

i would disagree since those numbers are within a range i guess.

OR means one or the other or both.

Suppose A={x,y,z} and B= {1,2,3}. Is x in A or B? Is x in A and B? Or and And are different.

Is 7 in x>2 or x<5? The answer is YES since 7 is in the set x>2

Is 7 in x>2 AND x<5? The answer is no since 7 is not in the set x<5


Why did you only test values between 2 and 5?

I didn't just test the numbers between 2 and 5, i also tested 6 and 1 and the result was bigger than 0. and I knew all the numbers that were smaller than 1 and bigger than 6 were going to result more than 0. So i didn't include those numbers. And numbers between 2 and 6 are not more than 0?
How did you test ALL numbers between 2 and 5?

I tested the numbers by substituting those numbers for X. if the result was less than 0 then it was part of the solution

What is the difference between x > 2 or x < 5 vs x > 2 and x < 5?

they look the same to me NO! This is where you are going wrong. For A AND B to be true we must have A is true and B is true. For A OR B to be true we must have that A is true or that B is true or that they are both true.

Can you give me 4 numbers between 2 and 5?

2,3,4,5, but i thought i shouldn't include the 2 and 5 since the equation has a < symbol 2 and 5 are NOT between 2 and 5. But 3.8 is between 2 and 5. sqrt(10), pi and 7/2 are also between 2 and 5. You wanted to test values less than 2 so you started with 1. For the record you could have tried 1.5894

It amazes me when students do math they forget everything they know about numbers in the real world. When you go to the store are the items always 1unit or 2 units or 3 units.... (I do not know the unit of your currency. For example in america the unit is dollar or $)? The answer is NO!. The prices are 6.05 or 43.65 or 245.78.... There are numbers other than the counting numbers

The original problem reduced to (x - 5) (x - 2) < 0. The left hand side is a product of two factors and this product is less than 0. To be able to solve this you must be able to answer this question. What can you tell me about the signs of these two factors?

 

[abel muroi]

You're fine to this point. Now you've got a product of two factors, which evaluates to a negative number. What must be true for a product of two values to be negative? The signs on the two values in the product must be opposite; one must be positive and the other must be negative. This gives you TWO cases:

. . . . .1) x - 5 < 0 and (not "or") x - 2 > 0

. . . . .2) x - 5 > 0 and (not "or") x - 2 < 0

(List these two cases in your homework solution!)

Now consider each of the two cases.

1) If x - 5 < 0, then x < 5. If x - 2 > 0, then x > 2. These two constraints together mean that 2 < x < 5.

2) If x - 5 > 0, then x > 5. If x - 2 < 0, then x < 2. But x cannot be BOTH greater than 5 and ALSO less than 2. So there is no x-value which works for these conditions.

Thus, only the first case "works".

Note that this method will work for messy numbers and for products of more than two factors, since you'll be able to split things into cases and consider each case logically.

hmm so basically, when a problem asks you to solve for x, they are just asking you to put the solution in the form 2 < x < 5 ?

 

[HallsofIvy]

hmm so basically, when a problem asks you to solve for x, they are just asking you to put the solution in the form 2 < x < 5 ?

If the problem asks you to solve an inequality then quite often the solutions set is one or more intervals.

 

[Steven G]

hmm so basically, when a problem asks you to solve for x, they are just asking you to put the solution in the form 2 < x < 5 ?

Yes, you have the right solution but I am concerned that you did not get it for the right reason.
You wrote x<5 OR x>2. This means -infinity < x < infinity. Note that it is very possible for a solution comes from x<5 OR x>2.

You should have gotten that x<5 AND x>2. This means that 2< x <5.

So why do we use AND? You quickly got (x - 5) (x - 2) < 0 which was good. Now when is a product of two factors less than 0? The answer is when one factor is positive AND one factor is negative.

This means x-5>0 AND x-2<0, or x-5<0 AND x-2>0

So we solve the two cases

Case 1: x-5>0 AND x-2<0 which means x>5 AND x<2. No numbers are greater than 5 AND less than 2.

Case 2: x-5<0 AND x-2>0 which means x<5 AND x>2. This means that 2 < x < 5.

You need to have a good understanding between AND and OR. Good luck!

 

[stapel]

hmm so basically, when a problem asks you to solve for x, they are just asking you to put the solution in the form 2 < x < 5 ?

No, they're asking you to solve for the values of x which result in a valid (that is, a true) statement. How you format that solution is your choice (unless they specify). The solution itself must follow logically from the original exercise statement. One cannot simply assume that the solution will always be of one particular type (or of any type at all, since some statements have no solution).

For instance, if the inequality, in your case, had been x² > 7x - 10, then the process would have been the same:

. . . . .x2 - 7x + 10 > 0

. . . . .(x - 5)(x - 2) > 0

Because the factors must have the same sign in order to have a positive result, you then would have done:

. . . . .x - 5, x - 2 > 0 or x - 5, x - 2 < 0

. . . . .1) x - 5, x - 2 > 0 then x > 5, x > 2
. . . . .for both to be true, x must be greater than 5

. . . . .2) x - 5, x - 2 < 0 then x < 5, x < 2
. . . . .for both to be true, x must be less than 2

. . . . .x cannot be both less than 2 and also greater than 5, so the solution
. . . . .is not one interval; instead, it is the two listed above.

. . . . .solution: x < 2 OR x > 5

Try considering the logic of what's going on. Trying to find a one-size-fits-all format for a perceived type of problem type is only going to lead to confusion and wrong answers.