Proofs

A proof is a series of claims that lead to a conclusion. Some proofs are conditional, which means that the claims can only be made under certain conditions. Click on a statement to see the proof

Double Inequality 1
if the following are true:
  • a > b
  • c = a
  • d = b

then c > d


Distributive Applied
(a + b) ⋅ (b ⋅ (-1)) = ((a ⋅ b) ⋅ (-1)) + ((b ⋅ b) ⋅ (-1))

Add Inverse
(b ⋅ a) + ((a ⋅ b) ⋅ (-1)) = 0

Equation a
((a + c) + d) + b = ((b + c) + a) + d

Equation
if b + c = 0, then (a + b) + (c + d) = a + d

Difference of Two Squares
(a ⋅ a) + ((b ⋅ b) ⋅ (-1)) = (a + b) ⋅ (a + (b ⋅ (-1)))

Reorder Terms 9
(b / c) ⋅ c = (b ⋅ c) / c

Multiplication Theorem
if not (c = 0), then (b ⋅ c) / c = b

When the Exponent is a Logarithm
b(logbx) = x

Converseofpowersubstitution
if bm = bn, then m = n

Inequality Greater Than
if b > a, then b + c > a + c

Simplify 4
(a + b) + (b ⋅ (-1)) = a

Inequality Problem
if x + 4 > 9, then x > 5

Inequality Problem 2
if y ⋅ 3 < 21, then y < 7

Inequality Problem 3
if y ⋅ (-2) < -16, then y > 8

Inequality Problem 4 2
if (x + 10) ⋅ (1 / (-5)) > 3, then x < -25

Inequality Problem 4
if (x + 10) / (-5) > 3, then x < -25

Subtract Same Sides Sub
if the following are true:
  • a = b
  • c = d

then a - c = b - d


Subtract Both Sides Pre 1
b + (c + (c ⋅ (-1))) = b + 0

Subtract Both Sides Pre 2
b + (c + (c ⋅ (-1))) = b

Subtract Both Sides Pre 3
if a = b + c, then a + (c ⋅ (-1)) = b + (c + (c ⋅ (-1)))

Add Number to Both Sides Pre
if a + 90 = 180, then 180 + (90 ⋅ (-1)) = a

Neg Nine Plus Nine Plus X Pre 1
((-9) + 9) + x = 0 + x

Neg Nine Plus Nine Plus X Pre 2
((-9) + 9) + x = x

Neg Nine Plus Nine Plus X
(-9) + (9 + x) = x

Substitute 6 Pre
if a = ((b + c) + d) + e, then a = (b + c) + (d + e)

Substitute 7 Pre
if b = d, then a + (b ⋅ (-1)) = a + (d ⋅ (-1))

Add Term to Both Sides 7 Pre
if a + b = c, then c + (a ⋅ (-1)) = b

Substitute First Term Pre
if a = e, then (a + b) + c = (e + b) + c

Substitution 16 Pre
if a + b = c, then (x + a) + b = x + c

Sum Equation Pre 1
(a + b) + c = (b + a) + c

Sum Equation Pre 2
if (a + b) + c = d, then (b + a) + c = d

Sum Equation Pre 3
if (a + b) + c = d, then a + c = d + (b ⋅ (-1))

Divide Both Pre
if d = e, then c / d = c / e

Multiply by 1
if a = b ⋅ 1, then a = b

Multiply by 1 Less
if a ⋅ 1 < b, then a < b

Add by 0 Less
if a + 0 < b, then a < b

Apply Associative Add
if b + c = d, then (a + b) + c = a + d

Apply Associative Multiply
if b ⋅ c = d, then (a ⋅ b) ⋅ c = a ⋅ d

Additive Inverse 2 Pre
c + (a ⋅ b) = c + (b ⋅ a)

Transitive Inequality
if the following are true:
  • a < b
  • c = a

then c < b


Greater Than Transitive Property Pre
if the following are true:
  • a > b
  • b > c

then c < a


Transitive Property of Inequality 3 Pre
if the following are true:
  • a > b
  • b = c

then c < a


Transitive Property of Inequality 4 Pre
if the following are true:
  • a > b
  • a = c

then b < c


Double Inequality 2
if the following are true:
  • a > b
  • a = c
  • b = d

then c > d


Inequality Change Pre
if the following are true:
  • a > b
  • c < 0

then b ⋅ c > a ⋅ c


Inequality Change
if the following are true:
  • a > b
  • c < 0

then a ⋅ c < b ⋅ c


Inequality Multiply by N1
if a < b, then a ⋅ (-1) > b ⋅ (-1)


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