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