Answer
The solutions are $a = 7$ or $a = 11$.
Work Step by Step
First, we isolate the absolute value expression on the left side of the equation. We do this by adding $4$ to both sides of the equation:
$|a - 9| = 2$
To get rid of the absolute value sign, we set up two equations where the expression within the absolute value sign is equal to both the negative and positive values of the constant on the right side of the equation:
$a - 9 = -2$ or $a - 9 = 2$
Add $9$ to each side of the equations:
$a = 7$ or $a = 11$
The solutions are $a = 7$ or $a = 11$.
To check our answers, we plug the values into the original equation.
Let's check $a = 7$:
$|7 - 9| - 4 = -2$
Evaluate what is in the absolute value sign:
$|-2| - 4 = -2$
Add $4$ to each side of the equation:
$|-2| = 2$
Take the absolute value:
$2 = 2$
The two sides of the equation are equal to one another; therefore, this solution is correct.
Let's check $a = 11$:
$|11- 9| - 4 = -2$
Evaluate what is in the absolute value sign:
$|2| - 4 = -2$
Add $4$ to each side of the equation:
$|2| = 2$
Take the absolute value:
$2 = 2$
The two sides of the equation are equal to one another; therefore, this solution is correct.
