Solve the equation x=4-|2x+1|

This type of equation involves a modulus, this the component enclosed in the straight lines, |2x+1|.

A modulus takes the absolute value of its contents, this means that regardless of the input you always have a positive value (or 0) as the output. e.g. |3|=3 and |-3|=3 also.

It’s helpful to remember that y=|x| looks like this:



This can be thought of as 2 separate lines:
i) y=x ii) y=-x.

Step 1: Rearrange the equation so that the modulus is on one side of the equation by itself.

x=4-|2x+1| rearranges to  
|2x+1|=4-x

By subtracting 4 and adding |2x+1| to both sides of the equation.

Step 2: Use the positive/negative property of the modulus to split the equation into 2 equations.

To take the positive form of the modulus, we remove the straight lines and multiply the contents by +1. This gives us the first equation:
(1): (2x+1)=4-x

To take the negative form of the modulus, we remove the straight lines and multiply the contents by -1. This gives us the second equation:
(2): -(2x+1)=4-x

N.B: (1) and (2) give us the equations for the 2 branches of our modulus graph (see above). We can visualise this graph by applying translation rules to y=|x| to form our y=|2x+1|.

Step 3: solve the 2 new equations to give 2 values of x.

(1): 2x+1=4-x      add x on both sides                      
3x+1=4           subtract 1 on both sides
3x=3             divide by 3 on both sides
x=1

(2): -(2x+1)=4-x  multiply out the brackets on LHS by multiplying by -1
-2x-1=4-x         add x on both sides
-x-1=4            add 1 on both sides
-x=5             multiply by -1 on both sides
x=-5

These are the solutions to the equation x=4-|2x+1|, x=-5 and x=1.