Solve Algebraic Equation: r > b/(sqrt2-1) = (sqrt2+1)b | Easy Explanation

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In summary, the conversation discusses a problem in probability and the use of algebra to solve it. The conversation also touches on rationalizing the denominator and the importance of understanding how to transform equations in the future.
  • #1
rwinston
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Hi

I am working through the superb book "50 challenging problems in probability." I have a slight problem with some of the algebra used in solving one of the problems. The algebra to solve is shown below:

[tex]
\frac{r}{r+b}\times\frac{r-1}{r+b-1}=\frac{1}{2}
[/tex]

Since, for b > 0:
[tex]
\frac{r}{r+b} > \frac{r-1}{r+b-1}
[/tex]

[tex]
\left(\frac{r}{r+b}\right)^2 > \frac{1}{2} > \left(\frac{r-1}{r+b-1}\right)^2
[/tex]

Thus
[tex]\frac{r}{r+b} > \frac{1}{\sqrt{2}} > \frac{r-1}{r+b-1}[/tex]

So (this is the part I have difficulty with):

[tex]r > \frac{b}{\sqrt{2}-1} = (\sqrt{2}+1)b[/tex]

I can't see how the two sides of the = sign can be transformed into each other, or how they are equivalent. Can anyone help with this?

Cheers
 
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  • #2
Divide both sides with [tex]\sqrt{2}+1[/tex] work the product in the denominator to yield one, and you'll see.
 
  • #3
I would have multiplied both sides by [itex]\sqrt{2}-1[/itex]!:rolleyes:
 
  • #4
But surely better than to verify, is to understand how to transform things like this in the future. It is rationalizing the denominator.
 

What is simple algebra?

Simple algebra is a branch of mathematics that deals with the manipulation and solving of equations involving symbols and variables.

What is an algebraic expression?

An algebraic expression is a mathematical phrase that contains numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

How do you solve a simple algebraic equation?

To solve a simple algebraic equation, you need to isolate the variable on one side of the equation by using inverse operations. This means that you perform the opposite operation to both sides of the equation until the variable is alone on one side.

What are the basic rules of algebra?

The basic rules of algebra include the commutative property, which states that the order of numbers can be changed without affecting the result, the associative property, which states that the grouping of numbers can be changed without affecting the result, and the distributive property, which states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.

What is the difference between an equation and an expression in algebra?

An equation is a mathematical statement that shows that two expressions are equal, while an expression is a mathematical phrase that does not have an equal sign and cannot be solved for a specific value.

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