Question

How do you write an absolute value inequality that represents all real numbers less than 3 units from 0?

Answer 1

Hint: We start solving the problem by assuming variable for all the real numbers satisfying the given condition. We then find maximum values of x on either side of 0 which are at a distance of 3 units from it. We then represent the obtained values by following the standard notations to get the required answer for the given problem.

Complete step-by-step answer:
According to the problem, we are asked to write an absolute value inequality that represents all real numbers less than 3 units from 0.
Let us assume the variable x to represent all the real values satisfying the given condition.
We know that the distance from –3 to 0 is equal to 3 units and also the distance between the 0 and 3 is also 3 units.
So, the values of x must lie between –3 and 3 which can be represent as $\left| x \right| < 3$ or $-3 < x < 3$.
$\therefore $ We have found the absolute value inequality that represents all real numbers less than 3 units from 0 as $\left| x \right| < 3$ or $-3 < x < 3$.

Note: Whenever we get this type of problems, we first assume the variables for the unknowns present in the problem to avoid confusion. Here we have taken modulus for x to represent the distance as we know that the distance is a positive quantity. We can also solve this problem by marking the numbers on the number line which also gives the similar answer. Similarly, we can expect problems to find the absolute value inequality of all real values that lies at a distance of 5 units from 5.