Maintaining Mathematical Proficiency
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The absolute value of a number is always zero or greater. Use this fact to examine the equations, what happens on the left-hand side?
See solution.
The exercise has three questions that have to be answered.
Since the absolute value of zero is zero we can ignore the absolute value in the equation which gives us x-y≠0 ⇔ x≠ y. Therefore, the possible values for x and y are all numbers where x≠ y.
As we examined in the previous question the absolute value of zero is zero, the sign will only change for negative numbers. Therefore, we can remove the absolute value on the left-hand side of the equation. x-y=0 If we move y to the right-hand side we get x=y. This means that x needs to be equal to y to satisfy |x-y|=0.
The condition states that the left-hand side should be a negative number, as it has to be below 0. This is not possible as an absolute value is defined as the possible value of the number. Even if the difference between x and y is negative, the result will be positive because of the absolute value. x-y < 0 Therefore, there are no values for x or y that satisfy |x-y|<0.