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. The core strategy hinges on a powerful, often overlooked principle: squaring both sides. And 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. 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 That's the part that actually makes a difference..
For non-negative numbers, the function f(x) = x² is strictly increasing for x ≥ 0. Because of this, for any inequality comparing two absolute values, squaring both sides is an equivalent transformation. The same logic applies to >, ≤, and ≥. This means if 0 ≤ a < b, then a² < b², and conversely, if a² < b² with a, b ≥ 0, then a < b. It preserves the truth of the inequality and eliminates the absolute value symbols, converting the problem into solving a polynomial inequality, typically quadratic Took long enough..
The general process is:
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- Still, , ax² + bx + c < 0). Solve the polynomial inequality using methods like finding roots and testing intervals. Expand and simplify the resulting expression into a standard polynomial inequality (e.g.Square both sides of the modulus inequality.
- Here's the thing — 3. Express the solution in interval notation or on a number line.
