Good explanation, but I do have a quibble with how you write "multiply by 6" and "multiply by (x+3)(x-3)": you write them superscripted, which looks like exponentiation. And if they remember the "exponent" visually before they remember the explanation, they might get confused. I've never seen anyone display multiplying both sides of an equation this way before.
Nice derivation. Note that at the end you do need to check that the resulting value of x is actually a solution because you multiplied with the denominators. When I do that, I prefer to always note the values of the variable that make the denominators 0 to remember that those cannot be solutions.
You bring up an interest point. Suppose the equation in the video is modified, changing the 8 on the RHS to 12. When the equation is solved, we get x = 3. So 3 looks like the answer, except then we see that when we plug 3 into the original equation, two of the expressions come out undefined. That would appear to be a problem. So 3 must not be a valid solution, but is that necessarily true? Is it possible the undefined parts cancel each other out? I mean, the imaginary parts of a calculation with complex numbers can cancel each other out, resulting in a real number, so why can't the same thing happen with undefined expressions? I am reminded of a problem from calculus: we want to integrate the function f(x) = 1/x from x = -1 to x = 2. The issue with this function is that it is undefined at x = 0, so the value of the integral is undefined (some would argue the area is infinite) from both -1 to 0 and from 0 to +2. However, the function clearly has pointwise symmetry about the origin, therefore the area of the function from -1 to 0 must equal the area from 0 to +1 - in a sense, the two areas are "equal infinities" that cancel each other out. Therefore, the area of f(x) from -1 to +2 must equal the area from +1 to +2, i.e. ln 2. Dissenting opinions from those who think this thinking is idiotic are of course welcome.
Or just short-cut the math by looking at the left-hand side as 2 fractions and multiply 2 times the x+3 and -4 times x-3, which equal 8. For the other question 1/2 is 3/6 and 1/3 is 2/6 so 5/6x equals 1/6.
lets rewrite this so it is a bit cleaner. -4 / (x + 3) + 2 / (x - 3) = 8 / (x² - 9) Now lets multiply both sides by (x- 3)(x + 3) -4 ( x -3 ) + 2 ( x + 3) = 8 Now we combine like terms and solve for x -4x + 2x + 12 + 6 = 8 -2x + 18 = 8 -2x + 18 - 18 = 8 - 18 -2x = -10 -2x / -2 = -10 / -2 x = 5 We must ALWAYS prove and check our work: [-4 / ( 5 + 3) ] + [2 / ( 5 - 3)] = [8 / (-9 + 5 * 5) [-4 / 8] + [2 /2] = [8/16] -1/2 + 1 = 1/2 1/2 = 1/2 The answer x = 5 is our only valid answer Sorry for the long answer, but if this was an exam question given by my old math teacher anything less would be marked as wrong answer.
You should really state the problem first. You want to solve the equation for x. Then you have to note that x is not 3 or -3 and so you can do the multiplication you propose,
What you are looking for is called the 'LOWEST common denominator', not the 'LEAST common denominator'. Basic English. The least common denominator would mean the RAREST denominator, which you would certainly NOT want to have to look for!
So next I suppose you're going to tell us the largest number that divides two integers isn't the *greatest* common factor, it's the *largest* common factor. The greatest common factor would be the best factor, right? Mathematicians never could use the English language correctly.
@@zanti4132 : 3×4=12, 12÷2=6 Is that: Three times four equals twelve. Twelve divided by two equals six. Three fours are twelve. Two into twelve is six. Three into four is twelve. Twelve by two is six. 🤔
That is not the meaning of "lleast" here. There are common denominators, and there is a least one of them. That is called a *"least common denominator."*
Good explanation, but I do have a quibble with how you write "multiply by 6" and "multiply by (x+3)(x-3)": you write them superscripted, which looks like exponentiation. And if they remember the "exponent" visually before they remember the explanation, they might get confused.
I've never seen anyone display multiplying both sides of an equation this way before.
Nice derivation. Note that at the end you do need to check that the resulting value of x is actually a solution because you multiplied with the denominators. When I do that, I prefer to always note the values of the variable that make the denominators 0 to remember that those cannot be solutions.
You bring up an interest point. Suppose the equation in the video is modified, changing the 8 on the RHS to 12. When the equation is solved, we get x = 3. So 3 looks like the answer, except then we see that when we plug 3 into the original equation, two of the expressions come out undefined. That would appear to be a problem.
So 3 must not be a valid solution, but is that necessarily true? Is it possible the undefined parts cancel each other out? I mean, the imaginary parts of a calculation with complex numbers can cancel each other out, resulting in a real number, so why can't the same thing happen with undefined expressions?
I am reminded of a problem from calculus: we want to integrate the function f(x) = 1/x from x = -1 to x = 2. The issue with this function is that it is undefined at x = 0, so the value of the integral is undefined (some would argue the area is infinite) from both -1 to 0 and from 0 to +2. However, the function clearly has pointwise symmetry about the origin, therefore the area of the function from -1 to 0 must equal the area from 0 to +1 - in a sense, the two areas are "equal infinities" that cancel each other out. Therefore, the area of f(x) from -1 to +2 must equal the area from +1 to +2, i.e. ln 2. Dissenting opinions from those who think this thinking is idiotic are of course welcome.
Or just short-cut the math by looking at the left-hand side as 2 fractions and multiply 2 times the x+3 and -4 times x-3, which equal 8.
For the other question 1/2 is 3/6 and 1/3 is 2/6 so 5/6x equals 1/6.
Always check your solution
lets rewrite this so it is a bit cleaner.
-4 / (x + 3) + 2 / (x - 3) = 8 / (x² - 9)
Now lets multiply both sides by (x- 3)(x + 3)
-4 ( x -3 ) + 2 ( x + 3) = 8
Now we combine like terms and solve for x
-4x + 2x + 12 + 6 = 8
-2x + 18 = 8
-2x + 18 - 18 = 8 - 18
-2x = -10
-2x / -2 = -10 / -2
x = 5
We must ALWAYS prove and check our work:
[-4 / ( 5 + 3) ] + [2 / ( 5 - 3)] = [8 / (-9 + 5 * 5)
[-4 / 8] + [2 /2] = [8/16]
-1/2 + 1 = 1/2
1/2 = 1/2
The answer x = 5 is our only valid answer
Sorry for the long answer, but if this was an exam question given by my old math teacher anything less would be marked as wrong answer.
You should really state the problem first. You want to solve the equation for x. Then you have to note that x is not 3 or -3 and so you can do the multiplication you propose,
Good one Mr. McLogan!
I think i would first observe x ≠ ±3
Interesting. I would have cross multiplied and then cancelled the denominators.
X= 5.
nice!
Interesting thanks, although a specific case. What would you have done if the two denominators didn’t multiply together to give the third?
What you are looking for is called the 'LOWEST common denominator', not the 'LEAST common denominator'.
Basic English.
The least common denominator would mean the RAREST denominator, which you would certainly NOT want to have to look for!
So next I suppose you're going to tell us the largest number that divides two integers isn't the *greatest* common factor, it's the *largest* common factor. The greatest common factor would be the best factor, right? Mathematicians never could use the English language correctly.
@@zanti4132 : 3×4=12, 12÷2=6
Is that:
Three times four equals twelve. Twelve divided by two equals six.
Three fours are twelve.
Two into twelve is six.
Three into four is twelve.
Twelve by two is six.
🤔
That is not the meaning of "lleast" here. There are common denominators, and there is a least one of them. That is called a *"least common denominator."*
@ davek6415 -- You wrote misinformation, and I reported it.
Lowest common denom. ? That's what we were taught in England, adding fractions at about age 8. The Americans just twist English enough to blur it.
bla bla bla bla bla. word diarrhoea
See a doctor then. 🤣