This is a continuation of Absolute Value.
Properties of the absolute value:
Non-negative: \( \vert x \rvert \geq 0 \) for all real values.
and for all real values.
Distance to origin: measures the geometric distance of the real number to the origin . Since distance is always positive, the absolute value of a number is always positive.
Thinking of absolute value as the distance from zero is also helpful when considering complex numbers . This distance from to the origin is given by the distance formula:
In fact, this is also the definition of the absolute value for a complex number :
1. Find all real values satisfying .
Solution 1: If , then we need to solve , which gives . This satisfies the original condition of , hence is a valid solution. If , then we need to solve , which gives or . This satisfies the original condition of , hence is a valid solution.
Solution 2: Using property 2, we obtain , or , which reduces to . This has solutions . We can verify that both of these are solutions.
2. Find all real values of satisfying ?
Solution: Let's approach this problem by considering the different regions. We have . Also, .
If then we have , or . This has roots , of which only the choice of satisfies . There is one solution in this case.
If , then we have , or . This has roots which are not real. Hence, there are no solutions in this case.
If , then we have , or . We check that the root is within this domain. Hence, there is one solution in this case.
If , then we have , or . This has no real root.
3. Prove the sub-additive property:
Solution 1: We apply the triangle inequality to the triangle with vertices , , and on the real number line. Then , implying .
If , then , so .
If , then , so .
If , then so . The same argument applies for .
If , then , so . The same argument applies for .