Solve the following equation. Show all your work.
(x)/(x-2)+(x-1)/(x+1)=-1
(x)/(x-2)+(x-1)/(x+1)=-1
clear the fractions by multiplying by the common denominator of (x-2)(x+1) and you have
(x)(x+1) + (x-1)(x-2) = -(x-2)(x+1)
x^2 + x + x^2 - 3x + 2 = -x^2 + x + 2
3x^2 - 3x = 0
x(x-1) = 0
x = 0,1
This one is ok, but you should always check for extraneous roots. In this case that would be x = -1,2 since the original equation would be undefined for those values.
please help me
Thank you so much!
x=0, or x=1
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Type in "x/x-2+(x-1)/x+1=-1" and click on step by step and just use that as your answer.
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Solve the following equation. Show all your work.
(x)/(x-2)+(x-1)/(x+1)=-1
(x)/(x-2)+(x-1)/(x+1)=-1
To solve this, we first need to find a common denominator for the two fractions on the left side of the equation. Since the denominators are (x-2) and (x+1), the common denominator is (x-2)(x+1).
Multiplying both sides of the equation by the common denominator, we get:
(x)(x+1) + (x-1)(x-2) = -1(x-2)(x+1)
Simplifying the left side:
x^2 + x + x^2 - 3x + 2 = -x^2 + x + 2
Combining like terms:
3x^2 - 3x = 0
Factoring out 3x:
3x(x-1) = 0
Now we have two solutions: x=0 and x=1.
However, we need to check whether either of these solutions would result in a denominator of 0, which would make the left side of the equation undefined.
Checking for x=0:
(x)/(x-2)+(x-1)/(x+1)=-1
(0)/(0-2)+(0-1)/(0+1)=-1
-1 = -1
So x=0 is a valid solution.
Checking for x=1:
(x)/(x-2)+(x-1)/(x+1)=-1
(1)/(1-2)+(1-1)/(1+1)=-1
-1/2 = -1
So x=1 is not a valid solution.
Therefore, the only solution to the equation is x=0.
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Anita can clean a typical pool in 8 hours. Chao can clean a typical pool in 6 hours. How long should it take Anita and Chao working together to clean a typical pool? Show all your work. Leave your final answer as an integer or reduced fraction.
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Anita can clean a typical pool in 8 hours. Chao can clean a typical pool in 6 hours. How long should it take Anita and Chao working together to clean a typical pool? Show all your work. Leave your final answer as an integer or reduced fraction.
Sure! Let x be the amount of time it takes for Anita and Chao to clean a typical pool working together.
Anita can clean a pool in 8 hours, so in one hour she can clean 1/8 of a pool. Similarly, Chao can clean 1/6 of a pool in one hour. Together, they can clean:
1/8 + 1/6 = 7/24
of the pool in one hour. Therefore:
7/24 x = 1
Multiplying both sides by the reciprocal of 7/24:
x = 24/7 hours, or approximately 3.43 hours.
Therefore, it would take Anita and Chao approximately 3.43 hours, or 3 hours and 26 minutes, to clean a typical pool working together.
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Simplify 4√6/√30 by rationalizing the denominator. Show your work.
To simplify 4√6/√30 and rationalize the denominator, we need to multiply both the numerator and denominator by the same expression that will eliminate the radical in the denominator. In this case, that expression is √30, since (√30)^2 = 30 and we can simplify the denominator by multiplying by √30/√30:
(4√6/√30) x (√30/√30) = (4√6√30)/(√30√30) = (4√6√30)/30
Simplifying the numerator:
4√(6x30) = 4√180 = 4√(4x45) = 4(2√45) = 8√45
Therefore, 4√6/√30 simplified and with a rationalized denominator is 8√45/30, which can be further simplified to 4√45/15.
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Simplify (2√5+3√7)2. Show your work. Justify each step.
To simplify (2√5+3√7)^2, we can use FOIL (First, Outer, Inner, Last) method to expand the square of the binomial expression.
FOILing, we get:
(2√5+3√7)(2√5+3√7) = (2√5)(2√5) + (2√5)(3√7) + (3√7)(2√5) + (3√7)(3√7)
Simplifying each term:
(2√5)(2√5) = 4(5) = 20
(2√5)(3√7) = 6√(5x7) = 6√35
(3√7)(2√5) = 6√(7x5) = 6√35
(3√7)(3√7) = 9(7) = 63
Combining like terms, we get:
20 + 12√35 + 63 = 83 + 12√35
Therefore, (2√5+3√7)^2 simplifies to 83 + 12√35.