Inequalities

Inequalities

Rules:

Adding/subtracting same value keeps inequality direction.
Multiplying/dividing by positive value keeps direction.
Multiplying/dividing by negative value reverses direction.

Sign chart method:

1. Bring all terms to one side.
2. Factorize.
3. Mark critical points on number line.
4. Test intervals.

Common patterns:

(x-a)(x-b) > 0 => x < min(a,b) or x > max(a,b)
(x-a)(x-b) < 0 => min(a,b) < x < max(a,b)

Modulus:

|x| < a => -a < x < a
|x| > a => x < -a or x > a
|x-a| = distance of x from a

Common traps:

• Reverse inequality when multiplying by a negative.
• Do not square both sides unless signs are controlled.