Solving Modulus-Equations

Question

So, I've come across this modulus equation in my textbook: $$ |2x-1|=|4x+9| $$

I looked at the solution to this equation and understand that in order for both sides to be equal, the quantities inside the brackets must either be the same or the negatives of each other.

The solution then uses the following theorem: if $|p| = b, b>0 \Rightarrow p = -b$ or $p = b $. $$ 2x-1 = -(4x+9) $$ or $$ 2x-1 = 4x+9 $$ and solves both linear equations to get $ x = -\frac{4}{3}$ or $ x = -5$

I then asked myself why the solution didn't bother to find $$ -(2x-1) = 4x+9 $$ or $$ -(4x-9) = -(2x-1) $$ and instead only found the two above. I then proceeded to calculate the above linear equations and got the exact same answers as above $ x = - \frac{4}{3} $ or $ x = -5$

I'd like to know why I achieved the same answers with this.

Answer 1

The equation $-(2x-1)=4x+9$ has the same solution as $2x-1=-(4x+9)$ as the only difference between these two equations is it has been multipled by $-1$.

Similarly $2x-1=4x+9$ has the same solution as $-(2x-1)=-(4x+9)$ for the same reason.

So to solve the original equation you need to solve only one out of the pair of equations $-(2x-1)=4x+9$ and $2x-1=-(4x+9)$ and then solve one out of the pair of equations $2x-1=4x+9$ and $-(2x-1)=-(4x+9)$. Which one you do from each pair doesn't matter as they have the same solution.

Answer 2

I drew a graph of $$ y = |4x+9| - |2x-1|. $$ Not to scale, for $x$ I made four squares equal $1,$ because the interesting $x$ values are $1/2$ and $-9/4.$ The graph is three lines joining up, but with different slopes (four times what I depicted). Had I checked the answer first, I would have moved it over a bit so as to see more of the negative $x$ axis.