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all correct except for last line:

-2x < 2
x > 2/(-2)
x > -1

jon

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instead of writing the word "integer" we use the symbol Z
instead of writing the word "rational" we use the symbol Q

Jon

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you don't have brackets around 10 + sqrt(16 - 9) so i do not know if the denominator of 8 is under everything, as you wrote or just the radical part... that is for you to clarify to yourself.

If it is everything, then is correct. Do not evaluate it...the question asks you to simplify NOT evaluate. Your answer is incorrect in that it is a rounded up version of a never ending decimal. That is why we write radicals... they are the EXACT number.

What we could do is:




but this is questionably NOT simpler than ... so leave it alone as as your answer.

jon.

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you now know that the gradient of the quoted equation is 6/4... or more simply 3/2.

As the line you are asked to find is parallel to this, we now know that its gradient is also 3/2. Put that into the standard equation of a straight line (y=mx+c) as the value of m.

We then need the value of c. To find that, you need to know x and y at a point, any point, which we DONT!: we are told that the line passes through the line x=3 which gives us the x-value but not the y-value.

So this question is worded badly if not incorrectly... there is no "equation" there are "equations"... in infinite number of them that are parallel to the quoted line and cross the line x=3

Please check the wording of the question. If it is as you wrote, then go back to your teacher and ask them what they are looking for, because the question has bad maths language.

Jon.

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your method is correct.

To get rid of the fractions in this question we will multiply every term by 12 and then divide the fractions:

(x/3) - (x/12) = -4
12*(x/3) - 12*(x/12) = -4*12
4x - x = -48
3x = -48
x = -48/3
x = -16

CHECK:
(-16/3) - (-16/12)
(-16/3) - (-4/3)
(-16/3) + (4/3)
(-16+4)/3
(-12)/3
-4 --> which is the correct answer, so we KNOW we are correct.

jon.

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it depends upon your level of maths knowledge. You do not state your level.

Look at the lessons on this website for Cramer's rule and possibly Gaussian Elimination, if you have come across matrices before.

Jon.

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any straight line is of the form y=mx+c.

If gradient is zero then that means m=0. We also know that x=1 when y=-8. So, we get -8=(0)(1)+c
--> c=-8

So, the equation is, starting from y=mx+c:
y=-8

jon

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P = 15.5(t+62) is the formula where P is in thousands and t=0 is 1980.

So,
1981 ==> t=1
1982 ==> t=2
1983 ==> t=3
1984 ==> t=4
...
1993 ==> t=13

So in 1993, we will have
P = 15.5(t+62)
P = 15.5(13+62)
P = 15.5(75)
P = 1162.5

So the population is 1162.5 thousands --> 1162.5*1000 --> 1,162,500

Back to 1980:
t=0 here so
P = 15.5(t+62)
P = 15.5(0+62)
P = 15.5(62)
P = 961
--> population is 961,000 in 1980

So the population increase from 1980 to 1993 is (1162500-961000)
201,500

jon.

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if the roots are (4+i)and (4-i), then we can say that:

x=(4+i) or x=(4-i)
hence
x-(4+i) = 0 or x-(4-i)=0

Working backwards we then say
[x-(4+i)][x-(4-i)] = 0
(x-4-i)(x-4+i) = 0








there you go.

jon.

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as it is written, 9-(-2)(-7) means:
9 minus (the product of -2 and -7)

(-2)*(-7) is +14

9-(-2)(-7)
9-14
-5

jon.

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i am assuming the equilibrium price is when supply=demand? If so, then use that...the 2 equations have to be equal:






So, factorise or use the quadratic formula to solve to get your answer(s).

jon.

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Assuming the log is base 10, then to remove the log we next to raise both sides to the base 10:



this is where the algebraic method and logs seem to me to be at odds to each other. Looking at the above line, we could do one of 2 approaches. Either divide the lefthand side first or move the 16x up to the righthand side.

Approach1:




Approach2:




so either x=0 or x-40=0
--> x=0 or x=40

This second approach seems to give us 2 answers whereas the first gave us just 1. However, putting x=0 into the original will give you an error, so the only solution is x=40... you must remember to check by substituting your answers back into the question, just to check.

jon.

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3(2-5)+14
3(-3)+14
-9+14
5

jon

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you cannot solve it...it isn't an equation...no equals sign. All you can do is simplify which will involve you factorising the numerator and denominator and then cancelling terms (if possible).


--> removed the negative sign to make the squared term positive, so it is easier in my head to factorise.





jon.

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look at the lesson i wrote on functions. It should help you.

Jon

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functions are VERY VERY simple things to use. They just look frightening because they are a more "professional" way of writing maths and to be honest a fair few maths teachers are rubbish at teaching too. Anyway...

All that f(x) =5x-1 means is that given the value "x", as quoted in f(x), then we have to multiply it by 5 and then subtract 1 from that.

eg f(2) means multiply it by 5 and then subtract 1.
--> f(x) =5x-1
--> f(2) = 5(2)-1
--> f(2) = 10-1
--> f(2) = 9

eg f(-6) means multiply it by 5 and then subtract 1.
--> f(x) =5x-1
--> f(-6) = 5(-6)-1
--> f(-6) = -30-1
--> f(-6) = -31

and so f(x) =5x-1 find f(a)
--> f(x) =5x-1
--> f(a) =5a-1
DONE

and finally
--> f(x) =5x-1
--> f(a-2) =5(a-2)-1
--> f(a-2) =5a-10-1
--> f(a-2) =5a-11
DONE

When asked to do questions like this, the above is ALL you do. I hope you see how easy it is now. Hope this helps.

jon.

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Just do 2 things in your head when you do maths:
1. "see" everything
2. take things in stages

OK... point 1 means that is, fully written, +

Point 2 means we are multiplying + by -. Then (as fractions do) we multiply the numerators and then also the denominators.

+ and - multiply to give -
2 and 3 (the numerators) multiply to give 6
3 and 1 (the denominators) multiply to give 3

So the answer is which simplifies to -2.

The aim is to just look at and do all this in your head and write and then -2.

====================================================
Now, before you ever multiply fractional terms together, what you should get into the habit of doing is doing any division first, to simplify terms. Since multiplication and division are opposites of each other, it does not matter which order you do these in, for example:

(5*3)/12 --> 15/12 --> 5/4

or
5*(3/12) --> 5*(1/4) --> 5/4

It is always PREFERABLE to divide first, so we keep the numbers as simple as we can when we finally multiply. In the above example we did 5*3 but in the second version we only had to do 5*1.

So, back to the question:
+

we can divide the two 3's, giving:
+

which gives -2, following the same processes in point 2 before.

Hope this helps.

jon

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you went wrong by forgetting the first term when you wrote the bracketed terms.

Factorising is all about converting something to "two" things multiplied.

Lets look at the starting terms:
we should know what this really means, since a lot of this is shorthand used by mathematicians. What we really have is:


or, further:


So to factor, we look at every term given... here we have just 2 terms, and we look for the most that is common to both.

So, start with the numbers. What is common? Answer is 6
Now the x's. What is common? Answer is xx
Now the y's. What is common? Answer is y
and finally the z's. what is common? Answer is z

So we shall take out 6xxyz from both terms.

What is left from the first term? Answer is 1*xxxxxx -->
And from the second term? Answer is 8yyyyyy -->

So we had 6xxyz(1xxxxxx - 8yyyyyy) which is correct but not very "professional" in its look. So we write it in a more condensed manner as:



And check: Multiply each of the terms in the bracket by and you should arrive back at the question. So get used to checking.

Now the skill is to not need the xxxxxxxyz notation. You should be able to look at and factorise it into in one line.

Now, knowing further maths skills we can spot something with the bracket:
can be written as



And this is of the form of which factorises. I cannot remember this standard result, but your textbooks should quote this for you. That will give you an even more factorised version of the answer.

jon.

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you need to build up the solution.

Even numbers: 2,4,6,8
Odd numbers: 1,3,5,7,9

The 5 digit number: XXXYY has to have 3 odds (the X's). So, lets look at just those:

Pick the first odd number. How many possibles are there? Answer 5 (1,3,5,7,9)
Now pick the second odd number. How many possibles are there now? Answer 4 (since one of them was picked to be the first number.
Similarly, the third odd number... there are 3 possibles.

So altogether, there are 5*4*3 --> 60 3-digit variations from 5 odd numbers.
Now for the evens. For a 2-digit number, the first of the 2 even numbers could be 1 of 4 numbers: answer = 4
The final even number must be one of the remaining 3 evens: answer = 3

So, altogether there are 4*3 --> 12 2-digit variations from 4 even numbers.

And now we combine them... we know how many odd permutations there are (60) and even permutations (12). Combined, there are 60*12 --> 720 permutations of 3 odd and 2 even.

As you can see... at no point did i use any "formula". I just worked it out from "first principles" and "common sense"

Hope this helps.
jon.

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this is testing a couple of things:

1. can you convert english to maths.
2. can you multiply out brackets.

the product of (3 less than twice x) and (2 more than the quantity 3 times x)

Product means 2 things multiplied...so we have to multiply the 2 brackets, once we have them as maths instead of english.

(3 less than twice x)
(3 less than 2x)
what would be 3 less than 10? Answer is 7. How did you get that? Well, we do the same in algebra... (2x-3)

(2 more than the quantity 3 times x)
(2 more than 3x)
(2+3x)
or (3x+2)

And now multiply the terms in the 2 brackets together:
(2x-3)(3x+2)



jon.

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bloody hell, what a horrible question! I am hoping for a lovely and simple answer though.





now 2a = 34/21
so a = 17/21

so we need to add a amount, which is , to make the left hand side the correct form for "completing the square". We cannot just add this term... we will also need to add the same amount to the other side too, to keep the equation "balanced":



The LHS can be re-written automatically as ... this is the whole point of doing what we did.



And now we need to work out the RHS.. use a calculator or do it longways, for practice...



and now scale up the other fraction to have the same denominator:



and hopefully you spot that 121/441 will be a lovely number when square rooted...
or
or
or
or
or

jon.

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x is zero as no other value would give the left hand equal to the right hand. Proof:

43x = x
43x-x=0
42x=0
so x = 0/42
--> x = 0

jon

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i do not understand what you mean by "solve an equation that has multiple sets" since you show no sets and yet you quote 3 equations rather than one, as "an equation" implies.

Please clarify what you mean. Also state your level of maths knowledge, as solving 3 equations in 3 unknowns (x,y,z here) is rather time consuming and there are specific methods utilising matrices.

jon.

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i am rusty on congruency but if ASA is a test for it (which i believe it is) then your solution is perfect.

Well done
Jon

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the differential gives you a turning point on a curve, either a max or a min (or a point of inflexion).

p(x)= -0.2x^2+300x-200
p'(x)= -0.4x+300 where p'(x) is the differential
p'(x)= -0.4x+300 = 0
0.4x = 300
x = 300/0.4
x = 750

p''(x) = -0.4 which is NEGATIVE. This is therefore a MAXIMUM point.

So profit is maximised at x=750

jon.

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the x-intercepts are where y=0. So put y=0 and then you need to, hopefully, factorise the remaining terms to find the solutions.

If you cannot factorise then go to the quadratic formula to find the answers.

jon

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do it in stages:
7a-7b-ac-bc
7(a-b)-ac-bc
7(a-b)-c(a-b)

and now lo and behold both halves have a similar term, (a-b), so take this out of both:
(a-b)(7-c)

there you go.

jon

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log2 8 = x

this is equivalent to asking "what power of 2 is needed to equal 8"? Answer is 3, since is 2*2*2 which is 8.

You need to become familiar with this form of question rather than having it written as . They are 2 opposite forms of the same problem just like 3-x=7 is just the same as asking x=3-7.

jon.

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2x > x + 10 > -x has 2 parts:

we need to work out and then also .

So,


And

which is just re-written... be careful with that inequality sign.

This becomes


So the answers are "x has to be greater than -5" but also, the first one says "x has to be greater than 10"

We cannot have both these answers since x=2 for example does not satisfy "x>10". So the one answer we are allowed to have is x>10 since anything greater than 10 satisfies both parts of the question.

jon.

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you carry on from here

jon.

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you can solve them anyway you fancy.

There are more regimented method using matrix theory... Gaussian Elimination or using determinants, as in Cramer's rule. But for a 4x4 matrix, this will kill you :-D

Good luck lol

Jon