Problem 51 - Calculus Single Variable

Question

If the set is given with absolute value signs, then write it without absolute value signs. If it is given without absolute value signs, then write it using absolute value signs. \(\\{w: w /(w+1)<0\\}\)

Step-by-Step Solution

Answer

The set with absolute values is \(\lvert w \rvert < \lvert w+1 \rvert\).

Step 1: Understand the Inequality

The inequality we have is \(\frac{w}{w+1} < 0\). The expression is negative when either the numerator or the denominator is negative, but not both.

Step 2: Identify Conditions for Negativity

For \(\frac{w}{w+1}\) to be negative, either \(w < 0\) and \(w+1 > 0\), or \(w > 0\) and \(w+1 < 0\).

Step 3: Solve Each Condition

Condition 1: \(w < 0\) and \(w+1 > 0\) implies \(w < 0\) and \(w > -1\). This inequality is impossible.Condition 2: \(w > 0\) and \(w+1 < 0\) implies \(w > 0\) and \(w < -1\). This inequality is also impossible.

Step 4: Consider Critical Points

Analyzing the inequalities at the critical points where \(w = 0\) or \(w = -1\) shows discontinuity; thus neither is in the solution.

Step 5: Conclude the Set Using Absolute Value

Since the only possible way for the function to be negative involves impossible sets, the inequality \(\frac{w}{w+1} < 0\) means \(\lvert w \rvert < \lvert w+1 \rvert\) to represent it with absolute values.

Key Concepts

Understanding Absolute Value

The concept of absolute value is crucial in mathematics. It represents the distance a number is from zero on the number line. This distance is always non-negative. When you see an expression like \(\lvert w \rvert\), it simply measures how far \(w\) is from zero, regardless of which direction.
Think of it as ignoring the sign of the number. So, if \(w\) is -3 or just 3, the absolute value is 3.
In an inequality context, absolute value helps in structuring expressions that indicate range without focusing on direction.
For instance, if you have \(|a| < b\), it means that \(a\) could be any number within the range of \(-b\) to \(+b\), which is very useful for defining the potential solutions of inequalities.

Solving Inequalities

Inequalities are expressions explaining how one value relates to another. They often use symbols like \( <, >, \leq, \geq\) to describe the relationship.
When solving inequalities, one useful strategy is to isolate the variable. For example, in \(\frac{w}{w+1} < 0\), we check when particles of the fraction are negative or positive.
Inequality solving can be tricky because multiplying or dividing both sides by a negative number reverses the inequality sign.
Always double-check these operations! Another thing to consider is domain restrictions. For example, \(w+1=0\) is undefined so we avoid this value.

Critical Points Analysis

Critical points are points in the domain of a function where the function’s value does not change direction or is undefined.
These points have a vital role in determining the behavior of a function, especially across intervals.
For the inequality \(\frac{w}{w+1} < 0\), critical points occur at \(w = 0\) and \(w = -1\).
At these points, the function changes or becomes undefined.
Analyzing critical points involves checking values around these points to assess where the function might change signs.
By understanding critical points, you gain insight into the structure of inequalities and can accurately define valid solution sets.