Write and absolute value equation that has the given solutions of x=3 and x=9
Given: x=3 and x=9 are solutions of the absolute value equation.
Note: There are multiple absolute value equations that will satisfy this problem. I will assume that the absolute value equation has to contain a linear expression (which does not contain any powers).
The general absolute value (linear) equation is of the form:
Where x is a variable, a and c are unknowns whose value we want to determine.
x=3 has to satisfy the equation, thus replace x in the absolute value equation by 3:
x=9 has to satisfy the equation, thus replace x in the absolute value equation by 3:
We now have two expressions that are equal to c, thus these two expressions have to be equal:
If the absolute value of two difference expression are equal, then the expressions have to be each other’s opposite:
Use distributive propert
Subtract a from each side of the equation:
Subtract 3 from each side of the equation:
Divide each side by −2:
We have determined the unknown a, then we still need to determine the unknown c. Equation that we found previously:
Replace a with −6:
Thus the unknowns are a=6 and c=3. Replace the unknowns by their value in the general absolute value equation |x−a|=c:
CONCLUSION: |x−6|=3 is an absolute value equation that has solutions x=3 and x=9.