Overview
The lecture covers solving inequalities involving the modulus (absolute value) function, teaching key interpretations, solution methods, and typical transformations used in coordinate geometry and calculus for competitive exams.
Modulus and Its Interpretation
- The modulus (mod) of a number x, written as |x|, represents its distance from 0 on the number line.
- If |x| ≤ a (where a > 0), x lies between –a and a.
- If |x| ≥ a (where a > 0), x is less than or equal to –a or greater than or equal to a.
- |x| is always non-negative (≥ 0); it can never be negative.
- Inequalities like |x| > –a or |x| ≥ 0 are true for all real x.
- Inequalities like |x| < –a have no solution, as a modulus can’t be negative.
Solving Modulus Inequalities
- For |x| ≤ a: Solution is x ∈ [–a, a].
- For |x| < a: Solution is x ∈ (–a, a).
- For |x| ≥ a: Solution is x ∈ (–∞, –a] ∪ [a, ∞).
- For |x| > a: Solution is x ∈ (–∞, –a) ∪ (a, ∞).
- If 0 < a < b, for a < |x| < b: Solution is x ∈ (–b, –a) ∪ (a, b).
- Exclude 0 from solutions if given |x| > 0 or 0 < |x| < a.](streamdown:incomplete-link)
Application to Functions
- These rules apply to any function inside modulus, not just x.
- For |f(x)| ≤ a, solution is f(x) ∈ [–a, a].
- For |f(x)| ≥ b, solution is f(x) ≤ –b or f(x) ≥ b.
Example Problem Breakdown
- To solve |x – 1| ≤ 3: Solution is x – 1 ∈ [–3, 3], giving x ∈ [–2, 4].
- For |x – 1| ≥ 7: Solution is x – 1 ≤ –7 or x – 1 ≥ 7, so x ≤ –6 or x ≥ 8.
- Combined, the union of intervals gives the general solution.
Complex Modulus Inequality Example
- For (|x| – 1)/(|x| – 2) ≤ 0:
- Let k = |x|.
- Solve k ∈ [1, 2) (since denominator can’t be zero).
- Final solution for x: x ∈ [–2, –1] ∪ 1, 2).
Key Terms & Definitions
- Modulus (|x|) — The non-negative value of x; the distance from zero.
- Inequality — A mathematical statement using <, ≤, >, or ≥ to compare expressions.
- Union (∪) — The combination of solution intervals.
- Intersection — The overlap of two intervals (where both conditions hold).
- Critical Points — Values where expressions inside modulus equal boundaries (e.g., ±a).
Action Items / Next Steps
- Practice solving modulus inequalities with different expressions and intervals.
- Review assignments or example problems related to modulus and inequalities.
- Prepare for upcoming chapters and practice previous years’ exam questions for this topic.