Question 1155747
<br>
You want an absolute value equation in the form<br>
{{{abs(x-c)=d}}}<br>
that has the solution set x=1/2 and x=-1/2.<br>
Interpret the equation as saying "the difference between x and c is d", allowing the difference to be either x-c or c-x.  On a number line, this is equivalent to saying that x and c are separated by a distance d.<br>
So for this particular problem, we are looking for numbers c and d for which both x=1/2 and x=-1/2 are the same distance d from c.<br>
If x=1/2 and x=-1/2 are the same distance d from c, then c has to be halfway between x=1/2 and x=-1/2; and the distance d is the distance from that c to either of the x values.<br>
Halfway between 1/2 and -1/2 is 0, so c is 0; and each of those x values is 1/2 unit from 0, so d is 1/2.<br>
ANSWER:
{{{abs(x-0) = 1/2}}}<br>
It is easy to show that x=1/2 and x=-1/2 are the only two solutions to that equation.<br>
Let's look at a more interesting example, with the two solutions being x=5 and x=13.<br>
Halfway between 5 and 13 is 9; and the distance from 9 to either 5 or 13 is 4.  So the equation is<br>
{{{abs(x-9)=4}}}<br>
That equation says the difference between x and 9 is 4; the two values for x are 9+4=13 and 9-4=5.<br>
Or...<br>
Solving that absolute value equation in the usual way, we have
{{{x-9=4}}}  or  {{{x-9 = -4}}}
{{{x = 13}}}  or  {{{x = 5}}}<br>