I need to know how to notate appropriately as well.

Solve p(x)=3

The x values which fulfill the conditions are: -5, -4, -3, -2, -1, 2.5


I am confused about when to use set notation {} or interval () and when I am supposed to list each number as opposed to listing two numbers and implying that all numbers that fall in between are included.

I know that [a,b] is inclusive of both a and b (a,b) excludes a and b, and (a,b] excludes a while including b.

Thank you.

Right. (0,1) is the set containing every number bigger than 0 and less than 1. So 0.5 is in (0,1) for example. (0,1) contains infinitely many numbers.

{0, 1} is the set containing just the numbers 0 & 1.

I'm on my phone, I'll write something when I get home.

Oh, this is classical mechanics, not quantum btw.

So if the x values which fulfill the conditions are: -5, -4, -3, -2, -1, 2.5

[-5,-4,-3,-2,-1] [union symbol]{2.5} ?

Nope. First, [x,y,z] isn't defined.

Second, is -2.7997534679 a solution? No? Then why are you giving an interval? You have 6 solutions. So write down a set with 6 elements.

That is my mistake. I explained it poorly. Visually what is actually happening is when p(x)=3, there is a horizontal line on the graph that starts at and includes (-5,3) and goes all the way to and includes (-1,3), then it does a little jump and comes back down to include (2.5,3).

I realize that when I gave that list, it implied those were the only numbers included. Now that I have re-explained it is [-5,-4,-3,-2,-1] [union symbol]{2.5} correct?

I also am uncertain (even if it no longer pertains) what you meant by [x,y,z] not being defined.

[x, z] denotes the set of all real numbers* between x and z, including x, z themselves.

{x, y, z} denotes the set containing the three elements x, y, z.

[x, y, z] denotes ??


* The notation can be used for any ordered set, although normally it is used for the reals.

Aaah, okay. SO:

[-5,-1] [union symbol]{2.5}
not
[-5,-4,-3,-2,-1] [union symbol]{2.5}

He means simply that [x,y,z] doesnt mean anything, referring to where you have written [-5,-4,-3,-2,-1]. If you mean all numbers between -5 and -1, you just write [-5,-1]. So assuming by your description you want to say "x is any number between -5 and -1 OR x is 2.5", you would write x \in [-5,-1] U {2.5}

EDIT: too slow

I appreciated seeing it written again, helps clarify it. Thanks.

No no.

[-5, -1] is an infinite set. Your solution is only 6 points!

I'm hoping you missed when I responded to you earlier, or else I'm really confused.

If the horizontal line includes its two endpoints, as you say, then yep this new answer is correct.

Yup I did miss that -- sorry.

[-5,-1]U{2.5} is right then.

Solving inequalities graphically. I'm assuming that means I am supposed to graph the inequality out and visually see what solves the inequality(i.e., find where the lines intersect)?

If that is the case I need a little refresher on how to graph an inequality. I think that I can proficiently solve the inequality algebraically, but I'm forgetting what it is I need to know about y=mx + b in order to solve graphically, if indeed that is what I am supposed to be concerned with at all.

x-(2x+4) > 0
x-2x - 4 > 0
-x-4 > 0

So at this point I have what looks to be a y=mx + b equation, and I know that my y intercept is at -4. If I was to graph this, I would simply draw a line through -4 and that descended over one down one over one down one etc. This would be my line for Y_1 correct?

x-(2x+4) > 0
x-2x - 4 > 0
-x-4 > 0
0 > x + 4

Now am I correct in assuming this is how I find out what Y_2 is? Again I would graph this by drawing a line through 4 on the y axis, ascending over one up one over one up one.

You had it right the first time. You have two functions, the first being -x-4 and the second being 0. If you want to solve graphically, you graph both and notice when the first is greater than the second. Here, I graphed the first function in red and the second in blue. Does that make sense?



You can also do it algebraically. You end up with x < -4, which is what you wanted. And here is the graph of that inequality:

Okay, so "0" is actually in mx + b form. y= 0/0 + 0 basically. So 0 is my y intercept and there is no slope.

Am I wrong when I think:
Y_1=-x-4
Y_2= 0

Is that unnecessary, will it confuse me or am I confusing another concept already?

I need help understanding if I'm doing something wrong here, and if not, how my answer reflects the books answer.

Solve the following inequality

-3(x-7) + 7x ≤ 4(x+6)
4x+21 ≤ 4x+24
from this point it seems I can get two answers
0 ≤ 3 or -3 ≤ 0

the answer in the book says: (-∞,∞)

I don't understand how my answer is the same as the interval notation answer (if it actually is), I also don't understand why if I am solving the inequality my variable vanishes. I was under the impression that when I am solving an equation/inequality, I am solving for the variable, so I thought my final answer should look like "x = y; x ≤ y; etc".

you got to 4x + 21 ≤ 4x + 24
and correctly then minus 4x from both sides to get
0 ≤ 3.

now for what values of x in the original form of the inequality does this hold true?


in general in inequalities you want to solve from a range of values for which the statement is true. both equalities and inequalities are mathematical statements that are true for certain values of a variable. an equation at the level of math you seem to be at is a statement that will in general only have one solution. inequalities will generally have a range of values that you have to solve for, but in many cases you can use the same algebra techniques (with nuances)

aaah yes. x can be any number. Got it.

Thanks

In interval notation are either of these just fine, or is one preferable:
(-1, ∞) U (-∞, -11/5) or (-∞, -11/5) U (-1, ∞)

Either is fine (i.e., they are both correct), but the latter is stylistically better since it's left to right.

Does anyone know what software/website alex used to make the graph in post #2343?