Question 230700
First let's review the idea of absolute value. The absolute value of a number is its distance from zero on the number line, without regard to direction.<br>
Your first inequality says that
{{{abs(x-1) < 1/2}}}
This says "the distance of x-1 from zero is less than 1/2". Now picture a number line and picture the numbers that would be within 1/2 of 0. Now, how do we describe these numbers in the form of inequalities. I hope the following makes sense. It is saying "x-1" is between -1/2 and 1/2":
{{{x-1 > -1/2}}} and {{{x-1 < 1/2}}}
Notice that the absolute values are gone. This is the key step in solving absolute value problems: learning how to remove the absolute values by writing equivalent inequalities (or equations). Now we just solve these inequalities by adding 1 to each side of each inequality:
{{{x > 1/2}}} and {{{x < 3/2}}}
This describes the solution set: "All numbers between 1/2 and 3/2 (not including 1/2 and 3/2)"<br>
{{{abs(x^2-1)<1/2}}}
This one says that the distance of {{{x^2-1}}} is less than 1/2. Using the same logic as above to rewrite it without absolute values we get:
{{{x^2-1 > -1/2}}} and {{{x^2-1 < 1/2}}}
Since there is no x term (just {{{x^2}}} terms, we'll isolate the squared terms"
{{{x^2 > 1/2}}} and {{{x^2 < 3/2}}}
Now we'll find the square root of each side:
{{{x > sqrt(1/2)}}} and {{{x < sqrt(3/2)}}}
Rationalizing the denominators we get:
{{{x > sqrt(2)/2}}} and {{{x < sqrt(6)/2}}}
So our solution to this problem is "all numbers between {{{sqrt(2)/2}}} and {{{sqrt(6)/2}}} (exclusive of {{{sqrt(2)/2}}} and {{{sqrt(6)/2}}}).