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horrorfan [7]
3 years ago
12

Can someone help me with algebra 2 with an explanation?

Mathematics
1 answer:
sammy [17]3 years ago
4 0

Answer: This took me a long time to type, hope you read it and find it helpful and easy to understand...

1. x=1

2. x=\frac{3}{2}

3. x=1

4. x=\frac{6}{5}

Step-by-step explanation:

1.

\frac{1}{x}-\frac{x-2}{3x}=\frac{4}{3x}

To make it easier to add / subtract fractions, we are looking for common denominators. I can see that if I move the fraction on the left with a denominator 3x to the right, I'll be able to add two equations with like denominator.

Let's add \frac{x-2}{3x}

\frac{1}{x}=\frac{4}{3x}+\frac{x-2}{3x}

Add the numerators and keep the same denominator.

\frac{1}{x}=\frac{4+x-2}{3x}

\frac{1}{x}=\frac{2+x}{3x}

To get rid of the denominator x on the left side, we can multiply by x.

(x)\frac{1}{x}=\frac{2+x}{3x}(x)

1=\frac{2+x}{3}

Now multiply by 3.

(3)1=\frac{2+x}{3}(3)

3=2+x

Subtract 2.

3-2=x\\1=x

The value of x is 1.

Proof.

\frac{1}{x}-\frac{x-2}{3x}=\frac{4}{3x}

\frac{1}{1}-\frac{1-2}{3(1)}=\frac{4}{3(1)}

1-\frac{-1}{3}=\frac{4}{3}

1+\frac{1}{3}=\frac{4}{3}

\frac{1*3+1}{3} =\frac{4}{3}\\\frac{4}{3}=\frac{4}{3}

I can't show the proof to the following problems because I have a 5000 character limit.

-----------------------------------------------------------------------------------------

2.

\frac{5x-5}{x^2-4x} -\frac{5}{x^2-4x}=\frac{1}{x}

Again, we have like denominators, therefore, we can simply subtract the numerators and keep the same denominator.

\frac{5x-5-5}{x^2-4x}=\frac{1}{x}

\frac{5x-10}{x^2-4x}=\frac{1}{x}

Again, we can multiply by x to get rid of the x in the denominator.

(x)\frac{5x-10}{x^2-4x}=\frac{1}{x}(x)

Do not distribute the x in the numerator yet because we're gonna eliminate it later.

\frac{(x)(5x-10)}{x^2-4x}=1

Factor the denominator.

\frac{(x)(5x-10)}{(x)(x-4)}=1

Simplify x's.

\frac{5x-10}{x-4} =1

Multiply by x-4 to get rid of the denominator.

(x-4)\frac{5x-10}{x-4} =1(x-4)

5x-10=x-4

Add 10

5x=x-4+10

Subtract x

5x-x=-4+10

Combine like terms;

4x=6

Divide by 4.

\frac{4x}{4}=\frac{6}{4}

x=\frac{6}{4}

Simplify by 2.

6/2 = 3

4/2 = 2

x=\frac{3}{2}

-----------------------------------------------------------------------

3.

\frac{x^2-7x+10}{x}+\frac{1}{x}=x+4

Same denominator, add numerators.

\frac{x^2-7x+10+1}{x}=x+4

\frac{x^2-7x+11}{x}=x+4

Multiply by x to get rid of the x.

(x)\frac{x^2-7x+11}{x}=(x+4)(x)

x^2-7x+11=x^2+4x

Subtract -x^2-4x

x^2-7x+11-x^2-4x=0

Combine like terms;

-11x+11=0

Subtract 11.

-11x=-11

Divide by -11

x=\frac{-11}{-11} \\x=1

-------------------------------------------------------------------------

4.

\frac{x^2+7x+10}{5x-30}+\frac{x}{x-6}=\frac{x^2-13x+40}{5x-30}

It's easier if you move the operation with denominator x+6 to the right side with negative sign and bring the operation on the right side to the left side with negative sign as well.

\frac{x^2+7x+10}{5x-30}-\frac{x^2-13x+40}{5x-30}=-\frac{x}{x-6}

Now, since we have the same denominators, we can simply subtract numerators.

\frac{x^2+7x+10-(x^2-13x+40)}{5x-30} =-\frac{x}{x-6}

Distribute the negative sign.

\frac{x^2+7x+10-x^2+13x-40}{5x-30} =-\frac{x}{x-6}

Combine like terms;

\frac{20x-30}{5x-30}=-\frac{x}{x-6}

Factor.

I can see that I can factor a 5 on both numerator and numerator. This will allow me to simplify them.

\frac{(5)(4x-6)}{(5)(x-6)} =-\frac{x}{x-6}

Simplify.

\frac{4x-6}{x-6}=-\frac{x}{x-6}

Multiply by x-6

(x-6)\frac{4x-6}{x-6}=-\frac{x}{x-6}(x-6)

This will simplify the denominators.

4x-6=-x

Add x and 6.

4x+x=6\\

Combine like terms;

5x=6

Divide by 5.

x=\frac{6}{5}

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