How to Solve Modulus Inequalities on Both Sides
Modulus inequalities, featuring absolute value expressions on both sides of the inequality sign, present a distinct challenge in algebra. Practically speaking, unlike standard inequalities or those with a single absolute value, they resist simple case-splitting due to the interplay between two distance-from-zero measurements. The core strategy hinges on a powerful, often overlooked principle: squaring both sides. This method leverages the fundamental property that absolute values are always non-negative, allowing us to transform the problem into a more familiar polynomial inequality without altering the solution set. Mastering this technique unlocks the ability to solve a wide range of problems efficiently and accurately.
The Golden Rule: Why Squaring Works
The absolute value of any real number, denoted |x|, represents its distance from zero on the number line and is therefore always greater than or equal to zero. When we have an inequality of the form |A| < |B|, |A| > |B|, |A| ≤ |B|, or |A| ≥ |B|, both sides are inherently non-negative. This is the critical insight Practical, not theoretical..
For non-negative numbers, the function f(x) = x² is strictly increasing for x ≥ 0. That's why, for any inequality comparing two absolute values, squaring both sides is an equivalent transformation. This means if 0 ≤ a < b, then a² < b², and conversely, if a² < b² with a, b ≥ 0, then a < b. The same logic applies to >, ≤, and ≥. It preserves the truth of the inequality and eliminates the absolute value symbols, converting the problem into solving a polynomial inequality, typically quadratic Easy to understand, harder to ignore..
The general process is:
- This leads to Expand and simplify the resulting expression into a standard polynomial inequality (e. , ax² + bx + c < 0). Square both sides of the modulus inequality. Still, g. 3. 2. 4. Solve the polynomial inequality using methods like finding roots and testing intervals. Express the solution in interval notation or on a number line.
