How To Solve Modulus Inequalities On Both Sides

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. Which means 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.

For non-negative numbers, the function f(x) = x² is strictly increasing for x ≥ 0. Which means the same logic applies to >, ≤, and ≥. 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. It preserves the truth of the inequality and eliminates the absolute value symbols, converting the problem into solving a polynomial inequality, typically quadratic.

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The general process is:

1. Square both sides of the modulus inequality.
2. Which means Expand and simplify the resulting expression into a standard polynomial inequality (e. g., ax² + bx + c < 0).
3. Solve the polynomial inequality using methods like finding roots and testing intervals.
4. Express the solution in interval notation or on a number line.

Step-by-Step