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Inequalities always true

(solution set = R) and modeling inequalities from word problems

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A Sets & Inequalities set, untimed

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Key formulas

12
  • UnionA ∪ B = { x : x ∈ A or x ∈ B }
  • IntersectionA ∩ B = { x : x ∈ A and x ∈ B }
  • Complement (relative to universe U)Ac = { x ∈ U : x ∉ A }
  • DifferenceA − B = { x : x ∈ A and x ∉ B }
  • Inclusion-exclusionn(A ∪ B) = n(A) + n(B) − n(A ∩ B)
  • Subset count of an n-element set2n subsets2n − 1 proper subsets2n − 1 nonempty subsets
  • Empty set ∅ is a subset of every set
  • Quadratic inequality ax2+bx+c>0solution keyed to root(s) and sign of a / discriminant Δ=b2−4ac (Δ>0 two roots, Δ=0 one, Δ<0 none)
  • Roots straddle k (one >k, one <k) for a>0f(k) < 0
  • Rational inequality f(x)/g(x) ≤ 0sign chart on numerator/denominator zeros (g(x) ≠ 0)
  • Absolute value|x − a| < b ⇔ a − b < x < a + b (b>0)
  • Inequality multiplicative propertymultiplying/dividing by a negative reverses the inequality

Shaky underneath?

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FoundationsAlgebra 2 foundationsStatistics and probability
Inequalities always true (solution set = R) and modeling inequalities from word problems — Sets & Inequalities