Comparing squares and solving inequalities

Comparing two squares

Comparing x² and y² is not always intuitive, as it depends on the sign of the numbers being compared. Here are some useful rules:

• If 0 ≤ a ≤ b, then a² ≤ b² (the square function preserves the order of positive numbers).
• If a ≤ b ≤ 0, then a² ≥ b² (the order is reversed for negative numbers).
• If a ≤ 0 ≤ b, no conclusion can be drawn without directly comparing the absolute values of a and b.

Example: let’s compare (-6)² and 4². Here, -6 ≤ 0 ≤ 4, so we compare |-6| = 6 with 4: since 6 > 4, we have (-6)² > 4², i.e. 36 > 16.

Solving x² ≤ a

To solve an inequality of the form x² ≤ k (where k > 0): x² ≤ k ⇔ -√k ≤ x ≤ √k

Example: x² ≤ 9 ⇔ -3 ≤ x ≤ 3, so S = [-3; 3].

Solving x² ≥ a

x² ≥ k (where k > 0) ⇔ x ≤ -√k or x ≥ √k

Example: x² ≥ 4 ⇔ x ≤ -2 or x ≥ 2, i.e. S = ]-∞; -2] U [2; +∞[.

Common pitfall

When solving x² ≤ 9, do not under any circumstances say “x ≤ 3” whilst forgetting the lower bound -3: half the solutions would be lost! Similarly, never divide an inequality by x without knowing whether x is positive or negative, as the direction of the inequality may then be reversed.