Question 1

Which of the following is the correct first step to solve the equation $$\frac{1}{x} - 1 = 2$$ for $$x$$?

Accepted Answer

Multiply both sides by $$x to $$eliminate the denominator.

Question 2

After multiplying both sides of $$\frac{1}{x} - 1 = 2 by$$ $$x$$, what is the resulting equation?

Accepted Answer

$$1 - x = 2x$$

Question 3

Solve for $$x$$: $$\frac{1}{x} - 1 = 2.$$

Accepted Answer

$$x = -\frac{1}{2}$$

Question 4

Which of the following best describes the process of clearing denominators in a rational equation?

Accepted Answer

Multiply both sides by the least common denominator (LCD) of all fractions.

Question 5

Given the equation $$\frac{2}{x} + 3 = 7$$, what is the value of $$x$$?

Accepted Answer

Question 6

A chemist needs to prepare a solution by diluting a concentrated acid. The equation $$\frac{1}{x} - 1 = 2$$ models the relationship between the concentration and the amount of water added. What does the solution $$x = -\frac{1}{2}$$ imply in this real-world context?

Accepted Answer

The solution is not physically meaningful because the amount of water cannot be negative.

Question 7

Which of the following is a common mistake when solving $$\frac{1}{x} - 1 = 2$$?

Accepted Answer

Forgetting to multiply both sides by $$x to $$clear the denominator.

Question 8

If you have the equation $$\frac{1}{x} - 1 = 2$$, and after clearing denominators you get $$1 - x = 2x$$, what is the next step to solve for $$x$$?

Accepted Answer

Add $$x to $$both sides to get $$1 = 3x.$$

Question 9

Which of the following equations is equivalent to $$\frac{1}{x} - 1 = 2$$ after all steps to isolate $$x$$?

Accepted Answer

$$x = -\frac{1}{2}$$

Question 10

A medical researcher is modeling the rate of a drug's absorption with the equation $$\frac{1}{x} - 1 = 2. If$$ $$x$$ represents the time in hours, what is the correct interpretation of the solution?

Accepted Answer

The model predicts a negative time, which is not possible, so the equation may not be appropriate for this scenario.

Question 11

Which of the following best explains why multiplying both sides of a rational equation by the variable in the denominator can introduce extraneous solutions?

Accepted Answer

Because multiplying by a variable can make the denominator zero, which is undefined.

Question 12

If you solve $$\frac{1}{x} - 1 = 2$$ and get $$x = -\frac{1}{2}$$, what should you do next?

Accepted Answer

Check the solution in the original equation to ensure it does not make the denominator zero.

Question 13

Which of the following is the correct sequence of steps to solve $$\frac{1}{x} - 1 = 2$$?

Accepted Answer

Multiply both sides by $$x$$, rearrange terms, and solve for $$x.$$

Question 14

A student claims that the solution to $$\frac{1}{x} - 1 = 2 is$$ $$x = 3. $$Which step did the student most likely make a mistake in?

Accepted Answer

They likely added $$1 to $$both sides and then took the reciprocal incorrectly.

Question 15

Which of the following is a valid check for the solution $$x = -\frac{1}{2} in $$the equation $$\frac{1}{x} - 1 = 2$$?

Accepted Answer

Substitute $$x = -\frac{1}{2}$$ into the original equation and verify both sides are equal.

Rational Equations: Methods and Examples

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