Inequality Rules — Complete Algebra Reference Sheet A complete reference for inequalities. Includes the basic inequality properties, the rule for flipping signs when multiplying by a negative, solving linear and quadratic inequalities, absolute value inequalities, and compound inequalities (AND/OR).
Inequality Symbols Strict vs Non-Strict
< , > (open dot) ; ≤ , ≥ (closed dot) <,\ > \text{ (open dot)};\ \le,\ \ge \text{ (closed dot)} < , > (open dot) ; ≤ , ≥ (closed dot) Basic Inequality Properties Addition Property
a < b ⟹ a + c < b + c a < b \implies a + c < b + c a < b ⟹ a + c < b + c Subtraction Property
a < b ⟹ a − c < b − c a < b \implies a - c < b - c a < b ⟹ a − c < b − c Multiplication by Positive
a < b , c > 0 ⟹ a c < b c a < b,\ c > 0 \implies a c < b c a < b , c > 0 ⟹ a c < b c Multiplication by Negative (FLIP)
a < b , c < 0 ⟹ a c > b c a < b,\ c < 0 \implies a c > b c a < b , c < 0 ⟹ a c > b c Division by Positive
a < b , c > 0 ⟹ a c < b c a < b,\ c > 0 \implies \tfrac{a}{c} < \tfrac{b}{c} a < b , c > 0 ⟹ c a < c b Division by Negative (FLIP)
a < b , c < 0 ⟹ a c > b c a < b,\ c < 0 \implies \tfrac{a}{c} > \tfrac{b}{c} a < b , c < 0 ⟹ c a > c b Transitive & Order Properties Transitive
a < b and b < c ⟹ a < c a < b \text{ and } b < c \implies a < c a < b and b < c ⟹ a < c Trichotomy
Exactly one: a < b , a = b , a > b \text{Exactly one: } a < b,\ a = b,\ a > b Exactly one: a < b , a = b , a > b Reciprocal Reversal
0 < a < b ⟹ 1 a > 1 b 0 < a < b \implies \tfrac{1}{a} > \tfrac{1}{b} 0 < a < b ⟹ a 1 > b 1 Reciprocal (Same Sign)
a < 0 < b ⟹ 1 a < 0 < 1 b a < 0 < b \implies \tfrac{1}{a} < 0 < \tfrac{1}{b} a < 0 < b ⟹ a 1 < 0 < b 1 Square Both Sides (positive)
0 < a < b ⟹ a 2 < b 2 0 < a < b \implies a^2 < b^2 0 < a < b ⟹ a 2 < b 2 Absolute Value Inequalities Less Than (AND)
∣ x ∣ < a ⟺ − a < x < a |x| < a \iff -a < x < a ∣ x ∣ < a ⟺ − a < x < a Less Than or Equal
∣ x ∣ ≤ a ⟺ − a ≤ x ≤ a |x| \le a \iff -a \le x \le a ∣ x ∣ ≤ a ⟺ − a ≤ x ≤ a Greater Than (OR)
∣ x ∣ > a ⟺ x < − a or x > a |x| > a \iff x < -a \text{ or } x > a ∣ x ∣ > a ⟺ x < − a or x > a Greater Than or Equal
∣ x ∣ ≥ a ⟺ x ≤ − a or x ≥ a |x| \ge a \iff x \le -a \text{ or } x \ge a ∣ x ∣ ≥ a ⟺ x ≤ − a or x ≥ a General Form
∣ f ( x ) ∣ < c ⟺ − c < f ( x ) < c |f(x)| < c \iff -c < f(x) < c ∣ f ( x ) ∣ < c ⟺ − c < f ( x ) < c Special Case
∣ x ∣ < 0 has no solution ; ∣ x ∣ ≥ 0 always true |x| < 0 \text{ has no solution};\ |x| \ge 0 \text{ always true} ∣ x ∣ < 0 has no solution ; ∣ x ∣ ≥ 0 always true Quadratic Inequalities Standard Approach
a x 2 + b x + c < 0 ⟹ find roots, test intervals ax^2 + bx + c < 0 \implies \text{find roots, test intervals} a x 2 + b x + c < 0 ⟹ find roots, test intervals Sign Chart
Mark roots on number line; test each interval \text{Mark roots on number line; test each interval} Mark roots on number line; test each interval Greater Than 0 (parabola up)
x < r 1 or x > r 2 x < r_1 \text{ or } x > r_2 x < r 1 or x > r 2 Less Than 0 (parabola up)
r 1 < x < r 2 r_1 < x < r_2 r 1 < x < r 2 Always Positive Test
Δ < 0 and a > 0 \Delta < 0 \text{ and } a > 0 Δ < 0 and a > 0 Always Negative Test
Δ < 0 and a < 0 \Delta < 0 \text{ and } a < 0 Δ < 0 and a < 0 Compound Inequalities AND (Conjunction)
a < x and x < b ⟺ a < x < b a < x \text{ and } x < b \iff a < x < b a < x and x < b ⟺ a < x < b OR (Disjunction)
x < a or x > b x < a \text{ or } x > b x < a or x > b Solving 'AND'
Apply same operation to all three parts \text{Apply same operation to all three parts} Apply same operation to all three parts Interval Notation (AND)
a < x < b ⟺ x ∈ ( a , b ) a < x < b \iff x \in (a, b) a < x < b ⟺ x ∈ ( a , b ) Interval Notation (OR)
x ∈ ( − ∞ , a ) ∪ ( b , ∞ ) x \in (-\infty, a) \cup (b, \infty) x ∈ ( − ∞ , a ) ∪ ( b , ∞ ) Rational Inequalities Standard Approach
P ( x ) Q ( x ) > 0 ⟹ find zeros of P , Q ; test intervals \frac{P(x)}{Q(x)} > 0 \implies \text{find zeros of } P,\ Q;\ \text{test intervals} Q ( x ) P ( x ) > 0 ⟹ find zeros of P , Q ; test intervals Exclude Denominator Zeros
Sign Change Points
Each zero of P or Q may flip the sign \text{Each zero of } P \text{ or } Q \text{ may flip the sign} Each zero of P or Q may flip the sign Common Mistake
Do NOT multiply by Q ( x ) if its sign is unknown \text{Do NOT multiply by } Q(x) \text{ if its sign is unknown} Do NOT multiply by Q ( x ) if its sign is unknown Famous Inequalities Triangle Inequality
∣ a + b ∣ ≤ ∣ a ∣ + ∣ b ∣ |a + b| \le |a| + |b| ∣ a + b ∣ ≤ ∣ a ∣ + ∣ b ∣ Reverse Triangle
∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a − b ∣ ||a| - |b|| \le |a - b| ∣∣ a ∣ − ∣ b ∣∣ ≤ ∣ a − b ∣ AM-GM Inequality
a + b 2 ≥ a b ( a , b ≥ 0 ) \frac{a + b}{2} \ge \sqrt{ab} \quad(a, b \ge 0) 2 a + b ≥ ab ( a , b ≥ 0 ) Cauchy-Schwarz
∣ u ⃗ ⋅ v ⃗ ∣ ≤ ∣ u ⃗ ∣ ⋅ ∣ v ⃗ ∣ |\vec{u} \cdot \vec{v}| \le |\vec{u}| \cdot |\vec{v}| ∣ u ⋅ v ∣ ≤ ∣ u ∣ ⋅ ∣ v ∣ Bernoulli's
( 1 + x ) n ≥ 1 + n x ( x ≥ − 1 , n ≥ 1 ) (1 + x)^n \ge 1 + nx \quad(x \ge -1,\ n \ge 1) ( 1 + x ) n ≥ 1 + n x ( x ≥ − 1 , n ≥ 1 ) 