Problem 59 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|0.5 x-0.75|<2$$

Step-by-Step Solution

Answer

Interval notation: \((-2.5, 5.5)\).

Step 1: Understanding the Absolute Value Inequality

We are given the inequality \(|0.5x - 0.75| < 2\). Absolute value inequalities of the form \(|A| < B\) mean that \(-B < A < B\). In our case, this implies \(-2 < 0.5x - 0.75 < 2\). This inequality needs to be split into two separate inequalities.

Step 2: Solve the First Inequality

Let's solve the inequality \(0.5x - 0.75 < 2\). Add 0.75 to both sides to isolate the term with \(x\): \(0.5x < 2 + 0.75\). Simplifying, we get \(0.5x < 2.75\). Divide by 0.5 to solve for \(x\): \(x < \frac{2.75}{0.5}\). Thus, \(x < 5.5\).

Step 3: Solve the Second Inequality

Now, let's solve \(-2 < 0.5x - 0.75\). Add 0.75 to both sides: \(-2 + 0.75 < 0.5x\). Simplifying, we get \(-1.25 < 0.5x\). Divide by 0.5 to solve for \(x\): \(\frac{-1.25}{0.5} < x\). Thus, \(-2.5 < x\).

Step 4: Combine the Solutions

Combine the results from Steps 2 and 3. We have \(-2.5 < x\) and \(x < 5.5\). This can be written as the double inequality \(-2.5 < x < 5.5\).

Step 5: Write in Interval Notation

The double inequality \(-2.5 < x < 5.5\) can be expressed in interval notation. This means that \(x\) is in the interval \((-2.5, 5.5)\). This is the solution to the original inequality.

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities can initially seem challenging, but breaking them down into simpler parts makes them quite manageable. An absolute value inequality such as \(|A| < B\) simplifies to a two-part inequality: \(-B < A < B\). The absolute value symbol \(|\ |\) implies the distance of a number from zero on a number line. Thus, the expression inside the absolute value should fall within this distance when compared to the number following the inequality symbol.

For solving \(|0.5x - 0.75| < 2\), we rewrite it as the compound inequality \(-2 < 0.5x - 0.75 < 2\). This tells us that the expression \(0.5x - 0.75\) lies between \(-2\) and \(2\). Handling these as separate, more straightforward inequalities helps simplify our problem, which is what we'll explore further in algebraic manipulation.

Interval Notation

Interval notation is a concise way of writing the set of all solutions to an inequality. Instead of using words or lengthy phrases, interval notation employs brackets \((\ )\) or square brackets \([\ ]\) to indicate whether the endpoints are included or excluded. A parenthesis means that the endpoint is not included, while a bracket indicates it is.

In our problem, the solution \(-2.5 < x < 5.5\) means \(x\) takes any value between \(-2.5\) and \(5.5\), not including these endpoints. This is written as \((-2.5, 5.5)\) in interval notation, where both \(-2.5\) and \(5.5\) are excluded from the solution set. The interval neatly encapsulates all possible values \(x\) can take, aligning with the inequality's requirements.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to solve equations or inequalities. It allows us to isolate the variable on one side and solve for its value. Let's break down how this applies to our inequality problem.

1. **Simplifying Equations**: Start by isolating the expression with \(x\) in the inequality \(0.5x - 0.75 < 2\). This involves basic steps like adding or subtracting numbers on both sides. For example, adding 0.75 to isolate \(0.5x\).
2. **Dividing**: Once isolated, divide by the coefficient of \(x\), here 0.5, to solve for \(x\). Simplifying gives \(x < 5.5\) from one inequality, and \(-2.5 < x\) from the second, combining neatly into a single range.

Algebraic manipulation simplifies the solving of inequalities, allowing clear steps to work through and reach solutions easily. This structured approach leads to finding the boundaries for \(x\), ultimately describing the solution set effectively in interval form.