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Adding the same number to both sides of an inequality generates an equivalent inequality. This equivalent inequality will have the same solution set and the inequality sign remains the same. Let x, y, and z be real numbers such that x< y. Then, the following conditional statement holds true.
If x< y, then x+z< y+z.
This property holds for the other types of inequalities.
Identity Property of Addition
Rewrite 0 as z-z
Commutative Property of Addition
- a-b=-(a+b)
Add parentheses
The last inequality can be rewritten using the biconditional statement. (y+z)-(x+z)> 0 ⇕ x+z< y+z Finally, because x< y, the property is obtained.
If x< y, then x+z< y+z.