Math / Algebra

QuestionGiven $y = \frac{x}{x-1}$ and $x > 1$ , which of the following is a possible value of $y$ ?
A. $-1.9$ B. $-0.9$ C. $0.0$ D. $0.9$ E. $1.9$

Studdy Solution

STEP 1

What is this asking? If $x$ is greater than 1, what are the possible values of $y$ when $y = \frac{x}{x-1}$ ? Watch out! Don't forget that $x$ cannot equal 1, because that would make the denominator zero, and we can't divide by zero!

STEP 2

1. Analyze the function
2. Explore extreme cases
3. Evaluate the options

STEP 3

Let's rewrite our function $y = \frac{x}{x-1}$ in a more insightful way!
We can add zero in the numerator as $x = x - 1 + 1$ , so $y = \frac{x - 1 + 1}{x-1}$ .

STEP 4

Now, we can split the fraction: $y = \frac{x-1}{x-1} + \frac{1}{x-1}$ .
This simplifies to $y = 1 + \frac{1}{x-1}$ .
Much better!

STEP 5

Since $x > 1$ , we know that $x - 1 > 0$ .
This means $\frac{1}{x-1}$ will always be positive.

STEP 6

Therefore, $y = 1 + \frac{1}{x-1}$ will always be greater than 1.

STEP 7

What happens when $x$ gets really, really close to 1?
Let's imagine $x = 1.000001$ .
Then $x - 1 = 0.000001$ , and $\frac{1}{x-1} = \frac{1}{0.000001} = 1,000,000$ .
So $y = 1 + 1,000,000 = 1,000,001$ ! $y$ gets super huge!

STEP 8

What if $x$ is gigantic, like $x = 1,000,001$ ?
Then $x - 1 = 1,000,000$ , and $\frac{1}{x-1} = \frac{1}{1,000,000} = 0.000001$ .
So $y = 1 + 0.000001 = 1.000001$ . $y$ gets really close to 1!

STEP 9

We know $y$ must be greater than 1.
Looking at our options, only one value is greater than 1: 1.9.

STEP 10

The answer is E. $y$ can be 1.9.

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QuestionGiven $y = \frac{x}{x-1}$ and $x > 1$ , which of the following is a possible value of $y$ ?
A. $-1.9$ B. $-0.9$ C. $0.0$ D. $0.9$ E. $1.9$

Studdy Solution

STEP 1

What is this asking? If $x$ is greater than 1, what are the possible values of $y$ when $y = \frac{x}{x-1}$ ? Watch out! Don't forget that $x$ cannot equal 1, because that would make the denominator zero, and we can't divide by zero!

STEP 2

1. Analyze the function
2. Explore extreme cases
3. Evaluate the options

STEP 3

Let's rewrite our function $y = \frac{x}{x-1}$ in a more insightful way!
We can add zero in the numerator as $x = x - 1 + 1$ , so $y = \frac{x - 1 + 1}{x-1}$ .

STEP 4

Now, we can split the fraction: $y = \frac{x-1}{x-1} + \frac{1}{x-1}$ .
This simplifies to $y = 1 + \frac{1}{x-1}$ .
Much better!

STEP 5

Since $x > 1$ , we know that $x - 1 > 0$ .
This means $\frac{1}{x-1}$ will always be positive.

STEP 6

Therefore, $y = 1 + \frac{1}{x-1}$ will always be greater than 1.

STEP 7

What happens when $x$ gets really, really close to 1?
Let's imagine $x = 1.000001$ .
Then $x - 1 = 0.000001$ , and $\frac{1}{x-1} = \frac{1}{0.000001} = 1,000,000$ .
So $y = 1 + 1,000,000 = 1,000,001$ ! $y$ gets super huge!

STEP 8

What if $x$ is gigantic, like $x = 1,000,001$ ?
Then $x - 1 = 1,000,000$ , and $\frac{1}{x-1} = \frac{1}{1,000,000} = 0.000001$ .
So $y = 1 + 0.000001 = 1.000001$ . $y$ gets really close to 1!

STEP 9

We know $y$ must be greater than 1.
Looking at our options, only one value is greater than 1: 1.9.

STEP 10

The answer is E. $y$ can be 1.9.

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QuestionGiven y=xx−1y = \frac{x}{x-1}y=x−1x​ and x>1x > 1x>1, which of the following is a possible value of yyy?A. −1.9-1.9−1.9 B. −0.9-0.9−0.9 C. 0.00.00.0 D. 0.90.90.9 E. 1.91.91.9

Studdy Solution

STEP 1

What is this asking? If xxx is greater than 1, what are the possible values of yyy when y=xx−1y = \frac{x}{x-1}y=x−1x​? Watch out! Don't forget that xxx *cannot* equal 1, because that would make the denominator zero, and we can't divide by zero!

STEP 2

1. Analyze the function2. Explore extreme cases3. Evaluate the options

STEP 3

Let's **rewrite** our function y=xx−1y = \frac{x}{x-1}y=x−1x​ in a more insightful way!We can **add zero** in the numerator as x=x−1+1x = x - 1 + 1x=x−1+1, so y=x−1+1x−1y = \frac{x - 1 + 1}{x-1}y=x−1x−1+1​.

STEP 4

Now, we can **split** the fraction: y=x−1x−1+1x−1y = \frac{x-1}{x-1} + \frac{1}{x-1}y=x−1x−1​+x−11​.This simplifies to y=1+1x−1y = 1 + \frac{1}{x-1}y=1+x−11​.Much better!

STEP 5

Since x>1x > 1x>1, we know that x−1>0x - 1 > 0x−1>0.This means 1x−1\frac{1}{x-1}x−11​ will *always* be **positive**.

STEP 6

Therefore, y=1+1x−1y = 1 + \frac{1}{x-1}y=1+x−11​ will *always* be **greater than 1**.

STEP 7

STEP 8

STEP 9

We know yyy must be greater than 1.Looking at our options, only one value is greater than 1: **1.9**.

STEP 10

The answer is E. yyy can be **1.9**.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGiven $y = \frac{x}{x-1}$ and $x > 1$ , which of the following is a possible value of $y$ ?
A. $-1.9$ B. $-0.9$ C. $0.0$ D. $0.9$ E. $1.9$

What is this asking? If $x$ is greater than 1, what are the possible values of $y$ when $y = \frac{x}{x-1}$ ? Watch out! Don't forget that $x$ cannot equal 1, because that would make the denominator zero, and we can't divide by zero!

1. Analyze the function
2. Explore extreme cases
3. Evaluate the options

Let's rewrite our function $y = \frac{x}{x-1}$ in a more insightful way!
We can add zero in the numerator as $x = x - 1 + 1$ , so $y = \frac{x - 1 + 1}{x-1}$ .

Now, we can split the fraction: $y = \frac{x-1}{x-1} + \frac{1}{x-1}$ .
This simplifies to $y = 1 + \frac{1}{x-1}$ .
Much better!

Since $x > 1$ , we know that $x - 1 > 0$ .
This means $\frac{1}{x-1}$ will always be positive.

Therefore, $y = 1 + \frac{1}{x-1}$ will always be greater than 1.

We know $y$ must be greater than 1.
Looking at our options, only one value is greater than 1: 1.9.

The answer is E. $y$ can be 1.9.