An absolute value is always a non-negative number because it measures the expression's distance from a midpoint on a number line.
|x-3|= 6This equation means that the distance is 6, either in the positive direction or the negative direction.
|x-3|= 6 ⇒ lx-3= 6 x-3= -6
We need to solve both of these cases so that we can find both values that satisfy the equation.
| x-3|=6
lc x-3 ≥ 0:x-3 = 6 & (I) x-3 < 0:x-3 = - 6 & (II)
lcx-3=6 & (I) x-3=-6 & (II)
(I), (II): LHS+3=RHS+3
lx= 9 x= -3
After solving an absolute value equation, it is necessary to check for extraneous solutions. To do this, we substitute the found solutions into the given equation and determine if a true statement is made.
Substituting 9 for x in the equation does result in a true statement, so x=9 is not an extraneous solution. Now let's check whether or not x=- 3 is extraneous.
Substituting (- 3) for x in the equation also does result in a true statement, so x=- 3 is not an extraneous solution. Therefore, the equation has two solutions.
