Absolute Value Equation

[Tarmac27]

Hello all,

Im having trouble with this absolute value equation. I have discovered that a common technique is considering what the graph of the absolute value function looks like. However, is there a way to solve this purely algebraicly?

[math]|x|+|x+3|+|x+5|=4x+20[/math]
Thanks.

 

[lex]

I.e. Write down the critical values (when each absolute expression equals 0). In this case, they divide the real number line into 4 regions. Write down an expression for each absolute value expression in each of the 4 regions (either plus or minus the expression, but now using brackets instead of absolute value signs) and solve the corresponding equations.
For each region check the potential solution actually lies in that region. If it does then it will be a solution of the original equation.

 

[Pedja]

Hello all,

Im having trouble with this absolute value equation. I have discovered that a common technique is considering what the graph of the absolute value function looks like. However, is there a way to solve this purely algebraicly?

[math]|x|+|x+3|+|x+5|=4x+20[/math]
Thanks.

You have to consider four different cases.

Case 1. [imath]x \in (-\infty,-5] \\ |x|=-x \\ |x+3|=-(x+3)\\ |x+5|=-(x+5)[/imath]

Case 2. [imath]x \in (-5,-3] \\ |x|=-x \\ |x+3|=-(x+3)\\ |x+5|=x+5[/imath]

Case 3. [imath]x \in (-3,-0] \\ |x|=-x \\ |x+3|=x+3\\ |x+5|=x+5[/imath]

Case 4. [imath]x \in (0,+\infty] \\ |x|=x \\ |x+3|=x+3\\ |x+5|=x+5[/imath]

Solve equation for each of these cases. The final solution is the union of the individual solutions.

 

[Tarmac27]

Ok sweet. Thanks guys

 

[lookagain]

You have to consider four different cases.

Case 4. [imath]x \in (0,+\infty] \\ |x|=x \\ |x+3|=x+3\\ |x+5|=x+5[/imath]

Solve equation for each of these cases. The final solution is the union of the individual solutions.

There needs to be a close parenthesis after the +oo.

I glanced at this case. Would your method arrive at lex's final answer?

 

[Pedja]

There needs to be a close parenthesis after the +oo.

I glanced at this case. Would your method arrive at lex's final answer?

That is a typo. Thank you for the correction. Yes , the final solution is [imath]-\frac{18}{5}[/imath] .