Problem 48 - Calculus Single Variable

Question

In Exercises \(48-51,\) 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. \(\\{x: x+5<|x+1|\\}\)

Step-by-Step Solution

Answer

The set can be rewritten as \(\{x: x < -3\}\).

Step 1: Understanding the Problem

We have the set defined as \(\{x: x+5<|x+1|\}\). This set is described using absolute value signs, and our task is to write it without them.

Step 2: Removing Absolute Value Using Definition

Recall that for any value \(y\), the absolute value \(|y|\) is defined as \(y\) if \(y \ge 0\), and \(-y\) if \(y < 0\). We apply this to \(|x+1|\).

Step 3: Consider Case 1: \(x+1 \geq 0\)

If \(x+1\ge0\), then \(|x+1| = x+1\). Therefore, the inequality becomes \(x+5 < x+1\). Simplifying gives \(4 < 0\), which is not possible. Hence, there are no solutions in this case.

Step 4: Consider Case 2: \(x+1 < 0\)

If \(x+1<0\), then \(|x+1| = -(x+1)\). The inequality becomes \(x+5 < -(x+1)\), which simplifies to \(x+5 < -x-1\). Simplifying further, we get \(2x < -6\), leading to \(x < -3\).

Step 5: Conclusion and Set Notation

From step 4, we identified that the set \(\{x: x+5<|x+1|\}\) can be written without absolute value signs as \(\{x: x < -3\}\). This is the solution.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that show a relationship of one quantity not being equal to another. They use symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). Solving inequalities involves finding the values that make these expressions true. Unlike equations, solving inequalities entails understanding how different expressions can compare to each other, either being less than or greater than. In the original exercise, we were given an absolute value inequality, meaning we not only needed to consider positive and negative cases, but also needed to rewrite the inequality without absolute values which complicate the direct comparison between terms.

To solve such an inequality, like the one given: \(x+5 < |x+1|\), we must break it down into two separate cases because absolute values can represent both positive and negative solutions. Hence:

• Consider when \(x+1\) is positive, leading to one interpretation of the inequality.
• Consider when \(x+1\) is negative, which flips the sign of the expression within the absolute value.

Breaking down these steps involves straightforward algebra often involving adding, subtracting, multiplying, or dividing terms while paying attention to reverse the inequality sign when necessary, especially after multiplication or division by a negative number. Each case in an absolute value inequality shows a different set of potential solutions.

Piecewise Functions

Piecewise functions are those that have different expressions based on varying conditions or intervals of the input variable. They play a crucial role when dealing with absolute values since absolute value expressions naturally split into different cases. This provides insights into how different sections of the domain behave regarding their output.

In terms of the inequality \(x+5 < |x+1|\), we treated it as a piecewise function by considering two cases:

• When \(x+1 \ge 0\): Here, since \(x+1\) is non-negative, the absolute value simplifies directly to \(x+1\).
• When \(x+1 < 0\): Since \(x+1\) is negative, the expression simplifies to \(- (x+1)\) because the absolute value of a negative is its positive form.

This piecewise consideration helps us figure out which side of a boundary we are looking at and how the function behaves in these distinct regions, leading to different feasible solutions in each scenario.

Set Notation

Set notation is a compact, efficient way of describing a collection of numbers or objects. With inequalities, set notation usually displays all possible solutions for an expression. It's like a mathematical shorthand that specifies conditions and solutions clearly.

In the original exercise, we used set notation to describe the solution of the inequality \(x+5 < |x+1|\). The notation began as \(\{x: x+5 < |x+1|\}\), and after solving the inequality, it simplifies to \(\{x: x < -3\}\).

This expression indicates that all values of \(x\) less than \(-3\) satisfy the initial condition of the inequality without the absolute values. The use of the colon \(:\) in set notation means "such that," instructing us on the condition every element \(x\) must meet. Using set notation allows us to easily communicate solutions and can be particularly helpful when dealing with complex inequalities, making mathematical communication precise and clear.