Problem 14 - Calculus Single Variable

Question

Write the set using interval notation. Use the symbol \(\cup\) where appropriate. \(\\{x:|x-2|<5\\}\)

Step-by-Step Solution

Answer

The interval notation is \((-3, 7)\).

Step 1: Understanding the Absolute Inequality

To solve the inequality \(|x - 2| < 5\), we recognize this as an absolute value inequality, which implies that the distance between \(x\) and 2 is less than 5.

Step 2: Splitting the Absolute Inequality

Convert the absolute inequality \(|x - 2| < 5\) into a compound inequality by splitting it into two linear inequalities: \(-5 < x - 2 < 5\).

Step 3: Solving the Compound Inequality

Solve each part of the compound inequality separately for \(x\). First, solve \(-5 < x - 2\) by adding 2 to both sides, which gives us \(-3 < x\). Second, solve \(x - 2 < 5\) by adding 2 to both sides, which gives \(x < 7\).

Step 4: Combining the Solutions

Combine the solved inequalities to write the range for \(x\). The solution \(-3 < x < 7\) describes the set of numbers that satisfy both conditions.

Step 5: Writing in Interval Notation

Convert the inequality \(-3 < x < 7\) into interval notation, which is \((-3, 7)\). This represents all the numbers \(x\) such that \(x\) is greater than -3 and less than 7.

Key Concepts

Absolute Inequalities

Absolute inequalities involve expressions within absolute value symbols. In mathematical terms, absolute value represents the distance from zero on the number line, disregarding any direction.
This is why you often see absolute inequalities like \(|x - a| < b\), where you are looking at the value of \(x\) that lies within a certain distance from \(a\). The inequality \(|x - 2| < 5\) implies the distance between \(x\) and 2 is less than 5. This is a common way to express a range, or set, of possible solutions for \(x\). By understanding the foundation of absolute inequalities, you gain insight into how these expressions hold constraints in equations.

Compound Inequalities

When dealing with absolute inequalities, splitting them into compound inequalities makes solving them easier. A compound inequality is essentially two separate inequalities that are combined into one. This allows a clearer path to solving for the variable.
In the example \(|x - 2| < 5\), you can break it down to \(-5 < x - 2 < 5\). Here, both inequalities need to be true simultaneously. Thus, you end up handling two linear inequalities:

• -5 < x - 2
• x - 2 < 5

Solving each individually confirms the range of possible values for \(x\). Hence, you transition from one complex inequality to simpler, more manageable parts.

Linear Inequalities

A linear inequality resembles a linear equation but instead of an equal sign, it uses inequality signs like <, >, ≤, or ≥. Solving linear inequalities involves similar steps as equations, such as adding, subtracting, multiplying, or dividing. However, one crucial rule is that multiplying or dividing by a negative number reverses the inequality sign.
For example, from the compound inequalities \(-5 < x - 2\) and \(x - 2 < 5\), you'd solve each by isolating \(x\):

• -5 < x - 2 ⇒ -3 < x
• x - 2 < 5 ⇒ x < 7

Combining these gives a linear inequality \(-3 < x < 7\). Representing it in interval notation, you get \((-3, 7)\). The interval shows all possible values of \(x\) that make the inequality true. Learning to work with linear inequalities is vital because they frequently appear in mathematical problems, indicating a spectrum of solutions rather than a solitary answer.