Question

Solve the following - Find all possible values for a

|a|=−a
|a–5|=5–a
a+5=−5−a
a+5=5+a
x−(−18) if x>−18

x−(−18), if x<−18

Answer 1

The definition of an absolute value:

`|a|=\left\{\begin{array}{ccc}a&for&a\geq0\\-a&for&a<0\end{array}\right`

========================================

`|a|=-a\to\boxed{a<0}\\----------------\\|a-5|=5-a\to|a-5|=\left\{\begin{array}{ccc}a-5&for&a\geq5\\5-a&for&\boxed{a<5}\end{array}\right\\-------------------\\a+5=-5-a\qquad\text{subtract 5 from both sides}\\a=-10-a\qquad\text{add}\ a\ \text{to both sides}\\2a=-10\qquad\text{divide both sides by 2}\\\boxed{a=-5}\\------------------------\\a+5=5+a\qquad\text{subtract}\ a\ \text{from both sides}\\5=5\qquad\text{TRUE}\\\boxed{\text{all real numbers}}\\----------------------\\x-(-18)=x+18`

`\text{If you mean absolute value}\\\\|x+18|=\left\{\begin{array}{ccc}x+18&for&x\geq-18\\-x-18&for&x<-18\end{array}\right\\\\--------------\\|x-(-18)|=x+18\ if\ x>-18\\--------------\\|x-(-18)|=-x-18\ for\ x<-18`

`\text{If you mean absolute value in third and fourth examples, then}\\\\|a+5|=\left\{\begin{array}{ccc}a+5&for&a\geq-5\\-a-5&for&a<-5}\end{array}\right\\\\|a+5|=-5-a=-a-5\to \boxed{a<-5}\\\\|a+5|=5+a=a+5\to\boxed{a\geq-5}`