Question

\(|[x /(x+1)]|<10^{-5}\) hold if (a) \(-10^{-5}<\mathrm{x}+1<10^{-4}\) (b) \(-(100001)^{-1}<\mathrm{x}<(99999)^{-1}\) (c) \((1 / 10000)<\mathrm{x}<1\) (d) \((99999)^{-1}<\mathrm{x}<(100001)^{-1}\)

Short answer

The correct answer is option (b), as the intervals \(x > \frac{1}{99999}\) and \(x < \frac{-1}{100001}\) hold for the given inequality.

Step-by-step solution

Rewrite the inequality without the absolute value

In this step, we'll rewrite the inequality \(|x /(x+1)|<10^{-5}\) without the absolute value. Given the quantity \(|x /(x+1)|\), we have two possible cases: the fraction is positive or it is negative. Rewrite the inequality for both cases: Case 1 (Positive): \(0 \le x/(x+1) < 10^{-5}\). Case 2 (Negative): \(-10^{-5} < x/(x+1) \le 0\). Now, we'll solve these inequalities separately.

Solve the inequality for the positive case

We are given the inequality \(0 \le x/(x+1)<10^{-5}\). We can first eliminate the \(x = 0\) scenario since this falls under our condition, and we're left with: \(0 \frac{10^{-5}}{1-10^{-5}}\) which simplifies to: \(x > \frac{1}{99999}\)

Solve the inequality for the negative case

We now consider the inequality \(-10^{-5} < x/(x+1) \le 0\). We can eliminate the inequality at 0, and we're left with: \(-10^{-5} -10^{-5}\) Now divide by \(1-10^{-5}\): \(x < \frac{-10^{-5}}{1-10^{-5}}\), which simplifies to: \(x < \frac{-1}{100001}\)

Compare the intervals with the given options

We found that \(x\) needs to be in one of the following intervals: \(x > \frac{1}{99999}\) or \(x < \frac{-1}{100001}\) Now let's compare these intervals to the given options: (a) \(-10^{-5}<\mathrm{x}+1<10^{-4}\): This is incorrect, as it doesn't match any of the above intervals we found. (b) \(-(100001)^{-1}<\mathrm{x}<(99999)^{-1}\): This interval matches the intervals we found, with \(x > \frac{1}{99999}\) and \(x < \frac{-1}{100001}\), so this option holds true. (c) \((1 / 10000)<\mathrm{x}<1\): This interval doesn't correspond with the intervals we found. (d) \((99999)^{-1}<\mathrm{x}<(100001)^{-1}\): This interval only matches the positive part of the intervals, neglecting the negative part, so it is partially correct, but option (b) is more accurate. From our analysis, the correct answer is option (b).

Key concepts

Absolute Value Inequalities

Absolute value inequalities are expressions that involve the absolute value of an algebraic variable or expression. The absolute value, denoted by two vertical bars (| |), measures the distance of a number from zero on the number line.In other words, the absolute value of a number is always non-negative.
To solve these inequalities, you need to consider two cases:

• Case 1: The expression inside the absolute value is non-negative (e.g., zero or positive).
• Case 2: The expression inside the absolute value is negative.

By analyzing both cases, you find possible solutions for each one.For example, if you have an inequality like \(|\frac{x}{x+1}| < 10^{-5}\),consider the situations when \(\frac{x}{x+1} \ge 0\) and \(\frac{x}{x+1} < 0\). This ensures all scenarios are addressed when the absolute value is removed.

Interval Notation

Interval notation is a concise way of expressing a range of values. It indicates the set of numbers that lie between a start and an end point on the number line.End points can be included or excluded from the set, represented using brackets.

• Parentheses \(( )\): Used to exclude an endpoint.
• Brackets \([ ]\): Used to include an endpoint.

For instance, the interval \((-10^{-5}, 10^{-4})\) means all numbers between \(-10^{-5}\) and \(10^{-4}\), without including these endpoints.
In the inequality solution, expressing results in interval notation helps simplify the presentation of a range of possible values for \(x\).
It allows for quick identification of the solution range without going into detailed expressions.

Algebraic Manipulation

In solving inequalities, algebraic manipulation is crucial.It involves rearranging and simplifying expressions to isolate the variable of interest.Consider breaking down complex expressions step by step to find solutions.Start by multiplying or dividing both sides of an equation by the same term, remembering to reverse the inequality when multiplying or dividing by a negative number.
For example, when solving the inequality \(0 < \frac{x}{x+1} < 10^{-5}\),we first multiply through by \((x+1)\) to clear the fraction. Then, algebraic operations like subtraction and factoring help simplify the expression further to isolate 'x'. Remember, every manipulation step must maintain the inequality's balance, leading to correct results.

Problem Solving Steps

Efficient problem solving involves a structured approach, ensuring that the solution is both accurate and logically derived.
Here are some key steps to solving absolute value inequalities:

• Identify the absolute value inequality: Analyze the given inequality and determine the form and possible cases for removal of the absolute value.
• Rewrite the inequality: Break down the inequality into two potential scenarios based on the absolute value definition.
• Solve each scenario separately: Use algebraic manipulation to solve the inequality for each case, ensuring all possible solutions are found.
• Combine and compare results: Collect possible solutions and express them in the appropriate format, such as interval notation. Compare solutions with given options or conditions to select the correct answer.

This structured procedure ensures that no detail is overlooked, and the solution is as comprehensive as possible.