SOLUTION: In the following problem a, b, c and d represent real numbers. Prove: If a = b and c < d, then a + c

SOLUTION: In the following problem a, b, c and d represent real numbers. Prove: If a = b and c < d, then a + c < b + d

Question 466359: In the following problem a, b, c and d represent real numbers.
Prove: If a = b and c < d, then a + c < b + d

Found 2 solutions by Theo, richard1234:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
a, b, c, and d are all positive numbers.
assume a = b
assume c < d
add a to both sides of the equation of c < d to get:
a + c < a + d
this is a valid statement based on the properties of arithmetic that state that you can add the same amount to both sides of an inequality and you will preserve the inequality.
now, since you are given that a = b, you can replace one of the a's in your equation with b and the equation will remain the same.
your equation of:
a + c < a + d becomes:
a + c < b + d
conversely:
a is equal to b
in the equation we just derived of a + c < b + d, you can replace b with a to get:
a + c < a + d
you can subtract a from both sides of this equation to get:
c < d
this proves that if a = b and c < d, then a + c < b + d
you are using some basic properties of arithmetic that prove this.
those properties are given and don't have to be proved.
they can be assumed to be true.
here's a reference
http://www.allaboutcircuits.com/vol_5/chpt_4/2.html
here's another reference
http://hotmath.com/hotmath_help/topics/properties-of-equality.html
here's another reference
http://hotmath.com/hotmath_help/topics/properties-of-inequality.html
the property that is used in the proof is the addition property.
with this last reference, the < symbol with a / superimposed on it means NOT smaller than, while the > symbol with a / superimposed on it means NOT greater than.
i can only show it as /> and /<.
the statement x /> x means that x is not greater than x, which is intuitively obvious because x = x.
read the references.
read the proof.
hopefully it will all make more sense to you after you're done.
an example may help clarify the concept.
let c = 4 and let d = 5
clearly c < d
let a and b both equal to 9.
start with c < d
substitute 4 for c and 5 for d to get:
4 < 5
add a to the left sides of this equation and add b to the right side of this equation.
you get:
a + 4 < b + 5
substitute 9 for a and 9 for b to get:
9 + 4 < 9 + 5
combine like terms to get:
13 < 14
you can also prove this in a reverse fashion by assuming that the statement is not true.
this would be:
if a = b and c < d, then a + c is not smaller than b + d
that means a + c is either equal to b + d or is greater than b + d.
let's make our statement to read:
if a = b and c < d, then a + c >= b + d
if we can disprove this statement, then the only other option is that it is smaller than b + d.
we start with:
a = b (given)
c < d (given)
our equation is:
a + c >= b + d
since b = a, we can replace b with a to get:
a + c >= a + d
if we subtract a from both sides of this equation, we get:
c >= d
but, we are given that c < d, so this equation must be false and we are left with the only other option left, which is that a + c < b + d.
once again, we can use an example:
let c = 4 and d = 5 and let a = 9 and b = 9.
we start with a + c >= b + d and we substitute to get:
9 + 4 >= 9 + 5 which becomes:
13 >= 14 which is false.
again, the contradiction based on the false assumption leading to the conclusion that:
if a = b and c < d, then a + c < b + d

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
To prove it in a more concise way, start by saying that if c < d, we can add equal quantities to both sides to get another true inequality. If we add a to both sides, we get

a + c < a + d

We can also replace a with b, and vice versa, because a and b are equal. Replace the a on the RHS side with b to get

a + c < b + d

Done.
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