Answer:
x = 8 and x = 2 are both valid solutions.
There are no extraneous answers.
Step-by-step explanation:
Given absolute value function:
[tex]|2(x-5)|+11=17[/tex]
To solve an equation containing an absolute value, isolate the absolute value on one side of the equation:
[tex]\implies |2(x-5)|+11=17[/tex]
[tex]\implies |2(x-5)|+11-11=17-11[/tex]
[tex]\implies |2(x-5)|=6[/tex]
Set the contents of the absolute value equal to both the positive and negative value of the number on the other side of the equation, then solve both equations.
Equation 1 (positive)
[tex]\implies 2(x-5)=6[/tex]
[tex]\implies \dfrac{2(x-5)}{2}=\dfrac{6}{2}[/tex]
[tex]\implies x-5=3[/tex]
[tex]\implies x-5+5=3+5[/tex]
[tex]\implies x=8[/tex]
Equation 2 (negative)
[tex]\implies 2(x-5)=-6[/tex]
[tex]\implies \dfrac{2(x-5)}{2}=\dfrac{-6}{2}[/tex]
[tex]\implies x-5=-3[/tex]
[tex]\implies x-5+5=-3+5[/tex]
[tex]\implies x=2[/tex]
Therefore, the solutions are x = 8 and x = 2.
Check if the solutions are valid by substituting them into the original equation:
[tex]\begin{aligned}x=8 \implies |2(8-5)|+11 & =17\\|2(3)|+11 & =17\\|6|+11 & =17\\6+11 & =17\\ 17 & = 17\end{aligned}[/tex]
[tex]\begin{aligned}x=2 \implies |2(2-5)|+11 & =17\\|2(-3)|+11 & =17\\|-6|+11 & =17\\6+11 & =17\\ 17 & = 17\end{aligned}[/tex]
Therefore, both solutions are valid and there are no extraneous answers.
Note: An extraneous solution is a solution that is produced by solving the problem, but is not a valid solution to the problem.
