Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=2 x+1\) must be less than 0.1 unit from 1.

Step-by-Step Solution

Answer

(a) \(|2x - 1| < 0.1\). (b) \(-0.05 < x < 0.05\).

Step 1: Express the problem in terms of absolute value inequality

The condition that the function \(f(x) = 2x + 1\) must be less than 0.1 units away from 1 can be represented as an absolute value inequality: \(|f(x) - 1| < 0.1\). This represents that the difference between \(f(x)\) and \(1\) is smaller than 0.1.

Step 2: Substitute the expression for \(f(x)\)

Substitute \(f(x) = 2x + 1\) into the inequality \(|f(x) - 1| < 0.1\). This gives us \(|2x + 1 - 1| < 0.1\), which simplifies to \(|2x| < 0.1\).

Step 3: Solve the absolute value inequality

The inequality \(|2x| < 0.1\) implies \(-0.1 < 2x < 0.1\). To solve for \(x\), divide each part of this compound inequality by 2: \(-0.1/2 < x < 0.1/2\), which simplifies to \(-0.05 < x < 0.05\).

Key Concepts

Inequalities

One of the foundational concepts in mathematics, especially in precalculus, is understanding inequalities. An inequality is a statement about the relative size or order of two numbers or expressions. Instead of saying two quantities are equal, inequalities define a range where these quantities can differ. This is symbolized by signs like ">", "<", "≥", or "≤".

In absolute value inequalities, such as the one in our example, we are often concerned with how far away a specific expression is from a particular value. An absolute value measures the distance between a number and zero on the number line, without considering direction. This is why inequalities involving absolute values are particularly useful when looking at a margin of error or deviation from a specified point. In practice, \(|a| < b\) translates to \(-b < a < b\). It allows us to express limitations or error margins in concrete terms, making them very useful in real-world situations.

Precalculus

Precalculus is a course that sets the stage for calculus by covering a breadth of mathematical concepts, including functions, complex numbers, and inequalities. Understanding inequality solutions within precalculus gives students a practical toolset for analyzing mathematical models. These models often indicate where values can exist and where they cannot, having crucial applications in fields like physics, engineering, and economics.

In the given exercise, we delved into the precalculus topic of absolute value to solve an inequality. This exercise illustrates how precalculus attempts to solidify foundational concepts that calculus will later build upon. By understanding how variables interact within inequalities, students learn to visualize restrictions on graphs, contributing to a deeper comprehension of mathematical relationships.

Functions

A function is essentially a rule that assigns to each input exactly one output. In our exercise, the function is expressed as \(f(x) = 2x + 1\). Functions are instrumental in mathematics because they describe how one quantity depends on another. When analyzing functions, it's crucial to understand what the function represents and how it behaves across different inputs.

In the context of our absolute value inequality, the function \(f(x)\) is graphed as a line. It's important to see how shifts and transformations, such as translating a graph by 1 or scaling it by 2, affect inequalities involving functions. Function analysis allows us to anticipate and understand changes and constraints depicted graphically or in equations.

• Always consider the domain: the set of possible inputs (\(x\) values).
• Look at the range: the set of possible outputs (\(f(x)\) values).
• Observe behavior: how does the function increase, decrease, or stay constant?

By mastering these aspects, we gain powerful insights into problem-solving and conceptual reasoning in calculus and beyond.