Question

Is anyone available to help with rational expressions and equations

Answer 1

Yes, I'll help you out.

Answer 2

thanks so much :)

Answer 3

(x-4) \ (x + 1) - (x - 2) \ (x - 1)

Answer 4

\[\frac{ x-4 }{ x+1 }-\frac{ x-2 }{ x-1 }\]
Here we are in front of a rational ecuation wich we'll have to apply some factoring to solve.

Let's begin by applying common denominator.
We need to multiply them in a way to make them equal, so let's do it.
We'll end up having:
\[\frac{ (x-4)(x-1)-(x-2)(x+1) }{ (x+1)(x-1) }\]
Now, we can see a square difference down in the denominator, so let's apply it:
\[\frac{ (x-4)(x-1)-(x-2)(x+1) }{ x ^{2}-1 }\]
Now, let's apply distributive on the denominator:
\[\frac{ (x ^{2}-x-4x+4)-(x ^{2}+x-2x-2) }{ x ^{2}-1 }\]
Let's get rid of the parenthesis:
\[\frac{ x ^{2}-x-4x+4-x ^{2}-x+2x+2 }{ x ^{2}-1 }\]
Now all we have to do is operate similar terms:
\[\frac{ -4x+6 }{ x ^{2}-1 }\]

And there we have it :)

Answer 5

thank you so much, you made that so simple! would you mind helping with another one?

Answer 6

Of course, just pass by.

Answer 7

thank you:) factor 27y^3 - 8x^6

Answer 8

\[27y ^{3}-8x ^{6}\]
Okay so first, we need to know a little about exponentials.
We know that 3^3 = 27, so let's plug that.
and 2^3 = 8, let's see how it looks:
\[3^{3}y ^{3}-2^{3}x ^{6}\]
Look on one side we have one that matches one of the properties:
\[(3y)^{3}-2 ^{3}x ^{6}\]
Okay, now, let's try to use another property to transformthat x^6
\[(3y)^{3}-2^{3}(x ^{3})^{2}\]
And there, i can't simplify it further :P

Answer 9

thank you so much for explaining so clearly! i really appreciate it!

Answer 10

i was working on another problem and how do you change 5x - x^2 to have the term with the exponent first?

Answer 11

i wish my teacher explained things as clearly as you do, i am grateful for your help!

Answer 12

Well, we call that "common factor"
\[5x-x ^{2}\]
What does the common factor say?
Simple! imagine if you took away an x from both terms.
the "5x" has an exponent of 1, wich means that it only has one x.
the "-x^2" has an exponent of 2, wich means it has 2 x's.
Let's apply it:
\[5x-x ^{2}\]
Would become:
\[x(5-x)\]
Notice that if I did distributive there, I would get the same result I factored :)

Answer 13

thank you...i was trying to make it the same as x^2 - 5x?

Answer 14

If you want to order it, it would be
\[-x ^{2}+5x\]
You'd change the result if you don't move it with it's corresponding sign.

Answer 15

thank you, that makes sense:)

Answer 16

are you available for help with another, or have i exceeded my limit?

Answer 17

Of course I'll help you with another one.
show me.

Answer 18

7 \ (x + 2) + 4 \ (3x + 6) = - 5\3

Answer 19

thank you :)

Answer 20

This one is a little more simple, but let's see:
\[\frac{ 7 }{ x+2 }+\frac{ 4 }{ 3x+6 }=-\frac{ 5 }{ 3 }\]
Okay so, let's see what the problem is asking us for, well the whole equation is equal to a number, so let's solve for x.
Let's apply common denominator on that sum.
\[\frac{ 7(3x+6)+4(x+2) }{ (x+2)(3x+6) }=-\frac{ 5 }{ 3 }\]
let's solve all those distributives.
\[\frac{ 21x+42+4x+8 }{ 3x ^{2}+6x+6x+12 }=-\frac{ 5 }{ 3 }\]
With them out of the way, let's operate similar terms:
\[\frac{ 25x+60 }{ 3x ^{2}+12x+12 }=-\frac{ 5 }{ 3 }\]
When we take Common factor, it doesn't necessarily have to be a variable, it can be a number, just to reduce those big numbers we have:
\[\frac{ 5(5x+12) }{ 3(x ^{2}+4x+4) }=-\frac{ 5 }{ 3 }\]
But... hey! we have a proportion, so i can simplify that -5/3
We'll end up with:
\[\frac{ 5x+12 }{ x ^{2}+4x+4 }=-1\]
Now let's pass that denominator multiplying to the other section of the equality:
\[5x+12=-1(x ^{2}+4x+4)\]
Let's do distributive with that -1 :
\[5x+12=-x ^{2}-4x-4\]
Let's pass that 5x and 12 to the other side so we can make everything equal to zero:
\[-x ^{2}-9x-18=0\]
All we have to do now is apply the general formula to solve that equation.

Answer 21

i am still struggling with this one, but i can easily follow your steps, so the answer would be x = -3 ?

Answer 22

It's a 2nd grade equation, it should pop out 2 solutions.
The equation is:
\[x=\frac{ -b \pm \sqrt{b ^{2}-4ac } }{ 2a }\]
Where, in our equation:
a=-1
b=-9
c=-18

Answer 23

Applying it, we will get two solutions, x=-3 and x=-6
the means that to solve that ecuation, x can take two values.
so we say:
S= (-5,-3)

Answer 24

thank you...i was just trying to follow how you did it vs. how i did and got a different answer

Answer 25

i got -7 when i worked through the problem

Answer 26

i did it differently from the beginning because i factored 3x + 6 to 3 (x+2) and then was working with 25 \ 3 (x + 2) which I set equal to -5/3...which became -5 (3x + 6) = 75 and then I got x = -7

Answer 27

Maybe i did something wrong, I'll double check.

Answer 28

thank you...i checked -7 and it works when i plug it back into the problem