Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 1<|x|<5 $$

Step-by-Step Solution

Answer

The solution is \((-5, -1) \cup (1, 5)\).

Step 1: Understand the Absolute Value Inequality

The given inequality is \(1 < |x| < 5\). This means that the absolute value of \(x\) is greater than 1 and less than 5. We need to solve the inequality \(|x| < 5\) and \(|x| > 1\) separately to find the solution for \(x\).

Step 2: Solve |x| < 5

The inequality \(|x| < 5\) can be rewritten as two separate inequalities: \(-5 < x < 5\).This means \(x\) can be any number greater than \(-5\) and less than \(5\).

Step 3: Solve |x| > 1

The inequality \(|x| > 1\) translates into two separate inequalities, \(x > 1\) or \(x < -1\).This means \(x\) can be any number greater than \(1\) or less than \(-1\).

Step 4: Combine the Solutions

To find the intersection of the solutions from Step 2 and Step 3, we combine the intervals.For \(-5 < x < 5\) and \(x > 1\), the overlap is \\(1 < x < 5\).For \(-5 < x < 5\) and \(x < -1\), the overlap is \\(-5 < x < -1\).Thus, combining these intervals, the solution to \(1 < |x| < 5\) is \((-5, -1) \cup (1, 5)\).

Step 5: Express Final Answer as Interval

Based on the intersections found in Step 4, the solution to the inequality is expressed as \[(-5, -1) \cup (1, 5)\].This represents all values of \(x\) which satisfy \(1 < |x| < 5\).

Key Concepts

Interval Notation

Interval notation is a way of representing a range of numbers along the number line. Through interval notation, we can concisely communicate a set of numbers that lie between two endpoints. For open intervals, like in our exercise, parentheses are used to denote that the endpoint values are not included in the set. For instance, \( (a, b) \) means all numbers greater than \(a\) but less than \(b\), excluding \(a\) and \(b\) themselves.
In our exercise, the solution \((-5, -1) \cup (1, 5)\) uses interval notation to show that \(x\) can be any number from \(-5\) to \(-1\) or from \(1\) to \(5\). The symbol \(() \cup ()\) indicates the union of two intervals, meaning all numbers that belong to either of the subsets are included in the solution.

Inequality Solving

Solving inequalities involves finding all possible values of a variable that satisfy the inequality statement. In mathematical expressions, inequalities express that one side is less than, greater than, or not equal to the other side. To solve inequalities involving absolute values, it's crucial to understand how absolute values work and apply that knowledge accordingly.
In our example, the inequality \( |x| > 1 \) is split into \( x > 1 \) or \( x < -1 \), and the inequality \( |x| < 5 \) is split into \( -5 < x < 5 \). This separation allows us to clearly explore the conditions under which each part of the inequality holds true and to combine these conditions to form the complete solution.

Absolute Value Concept

The absolute value of a number is the distance between that number and zero on the number line, always expressed as a positive quantity or zero itself. It's symbolized as \(|x|\) and relies on the property that the distance is always non-negative.
In an inequality, the absolute value represents the outcomes from a defined center point—in this case, zero. For example, \( |x| < 5 \) requires that \(x\) be less than 5 units away from zero on either side, covering all numbers between \(-5\) and \(5\). Similarly, \( |x| > 1 \) indicates that \(x\) lies beyond 1 unit away from zero, resulting in two parts of the number line being relevant: to the left of \(-1\) and to the right of \(1\).

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators that represent a particular value or equation. They don't include equality or inequality signs by themselves unless part of a condition, formation, or equation statement.
When dealing with expressions like \(|x| < 5\), understanding mathematical syntax and operations is essential. Rewriting \( -5 < x < 5 \) correctly interprets this expression by transforming inequality into a more tangible solution set. Expressions inform both the path of solving a problem and the kind of manipulation needed, be it through algebraic transformation or logical considerations.