Hi, I have a test tomorrow and our teacher gave us a huge review worksheet. I am still not sure how to do some of these so I would appreciate if you could do the following problems out in all steps. I tried to pick some from each section.
- Thanks :)
1. 7x+2/6x + 2x-11/6x
2. x^2/x+5 - 25/x+5
3. 5/6x-2y - 3/9x-3y
4. x/2x-4 - 2/x^2-2x
5. x/x+3 - 5/x+2
6. x-4/y^2-x-2 + x-7/x+1
Sorry for number six, the bottom of the first fraction is an x^2 NOT a y^2 very sorry.
In each case, you should have brackets around what I assume is the numerator and the denominator.
e.g. in #1 (7x+2)/(6x) + (2x-11)/(6x)
since you have a common denominator already, just add the tops
= (9x-9)/(6x)
= (3x-3)/(2x) after dividing top and bottom by 3
#2 becomes (x^2-25)/(x+5) since we again have a common denominator
= (x+5)(x-5)/(x+5) , the top was a difference of squares
= x-5
in #3 factor the two denominators to
2(3x-y) and 3(3x-y)
in the second fraction you can cancel the 3's
and then your common denominator is 2(3x-y)
etc
in #4
x/(2x-4) - 2/(x^2-2x)
= x/[2(x-2)] - 2/[x(x-2)]
common denominator will be 2x(x-2)
so we have
x^2/[2x(x-2)] - 4/[2x(x-2)]
= (x^2 - 4)/[2x(x-2)]
= (x+2)(x-2)/[2x(x-2)]
= (x+2)/(2x)
do #5 and #6 the same way
in #6 the first denominator factors to
(x-2)(x+1), which then becomes your common denominator.
Let me know what you get.
3(x-2)>4(2x+11)
Of course! I'd be happy to help you with the problems. Let's solve each one step by step:
1. 7x + 2 / 6x + 2x - 11 / 6x
To simplify this expression, we need to combine the fractions into a single fraction.
First, we have (7x + 2) / (6x). To make it easier to combine the fractions, we can express 2 as (2x / 6x):
(7x + 2x) / (6x).
Next, we have (-11) / (6x). We can multiply the numerator and denominator by -1 to change the sign of the fraction:
(-1 * 11) / (6x).
Combining the fractions, we get:
(7x + 2x - 11) / (6x).
Simplifying further, we have:
9x - 11 / 6x.
2. x^2 / (x + 5) - 25 / (x + 5)
In this problem, we have two fractions. To combine them, we need a common denominator, which in this case is (x + 5).
For the first fraction, x^2 / (x + 5), the denominator is already (x + 5).
For the second fraction, 25 / (x + 5), we can rewrite it as (25 / 1) / (x + 5) to make it a division of fractions.
Now, we have:
(x^2 / (x + 5)) - ((25 / 1) / (x + 5)).
To subtract these fractions, we need a common denominator, which is already (x + 5).
Combining the fractions, we get:
(x^2 - 25) / (x + 5).
3. 5 / (6x - 2y) - 3 / (9x - 3y)
To combine these fractions, we need a common denominator. In this case, it is (6x - 2y)(9x - 3y).
For the first fraction, 5 / (6x - 2y), we can multiply the numerator and denominator by (3x - y), which is the missing factor from the denominator:
(5 * (3x - y)) / ((6x - 2y)(3x - y)).
For the second fraction, 3 / (9x - 3y), we can multiply the numerator and denominator by 2, again to create the missing factor:
((3 * 2)) / ((9x - 3y)(2)).
Combining the fractions, we get:
(5 * (3x - y) - (3 * 2)) / ((6x - 2y)(3x - y)(2)).
Simplifying further, we have:
(15x - 5y - 6) / ((6x - 2y)(3x - y)(2)).
4. x / (2x - 4) - 2 / (x^2 - 2x)
To combine these fractions, we need a common denominator. In this case, it will be (2x - 4)(x^2 - 2x).
For the first fraction, x / (2x - 4), we can leave the numerator and denominator as they are.
For the second fraction, 2 / (x^2 - 2x), we can rewrite it as (2 / 1) / ((x)(x - 2)) to create the missing factors.
Combining the fractions, we get:
(x - (2 / 1)) / ((2x - 4)(x)(x - 2)).
Simplifying further, we have:
(x - 2) / ((2x - 4)(x)(x - 2)).
5. x / (x + 3) - 5 / (x + 2)
To combine these fractions, we need a common denominator. In this case, it will be (x + 3)(x + 2).
For the first fraction, x / (x + 3), we can multiply the numerator and denominator by (x + 2) to create the missing factor:
(x * (x + 2)) / ((x + 3)(x + 2)).
For the second fraction, 5 / (x + 2), we can multiply the numerator and denominator by (x + 3):
(5 * (x + 3)) / ((x + 2)(x + 3)).
Combining the fractions, we get:
((x * (x + 2)) - (5 * (x + 3))) / ((x + 3)(x + 2)).
Simplifying further, we have:
(x^2 + 2x - 5x - 15) / ((x + 3)(x + 2)).
Simplifying the numerator, we get:
(x^2 - 3x - 15) / ((x + 3)(x + 2)).
6. (x - 4) / (y^2 - x - 2) + (x - 7) / (x + 1)
To add these two fractions, we need a common denominator. In this case, it will be (y^2 - x - 2)(x + 1).
For the first fraction, (x - 4) / (y^2 - x - 2), we can leave the numerator and denominator as they are.
For the second fraction, (x - 7) / (x + 1), we can multiply the numerator and denominator by (y^2 - x - 2):
((x - 7) * (y^2 - x - 2)) / ((x + 1)(y^2 - x - 2)).
Combining the fractions, we get:
((x - 4) * (x + 1) + (x - 7) * (y^2 - x - 2)) / ((y^2 - x - 2)(x + 1)).
Expanding and simplifying the numerator, we have:
(x^2 - 3x - 4 + xy^2 - 3y^2 + 8y + 4x - 7) / ((y^2 - x - 2)(x + 1)).
Simplifying further, we get:
(x^2 + xy^2 + x - 3x - 4x - 3y^2 + 8y - 7) / ((y^2 - x - 2)(x + 1)).
Combining like terms in the numerator, we get:
(x^2 + xy^2 - 6x - 3y^2 + 8y - 7) / ((y^2 - x - 2)(x + 1)).
These are the step-by-step solutions to the problems you provided. Let me know if you have any further questions!