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

We can start by rearranging the equation such that the modulus (also called the absolute value) is the only component on the right-hand-side of the equation.

This gives us: x-4=-|2x+1| (by subtracting 4 from each side) and we shall call this equation (1)

The modulus of a number may be thought of as its distance from zero e.g. |x|=x for a positive x, |x|=-x for a negative x.

Therefore, equation (1) can be written as two separate equations:

x-4=-(2x+1)   (2)   &   x-4=(2x+1)   (3)

We can now rearrange equation (2) and solve for x as follows:

x-4=-2x-1   (we can then plus 2x to each side)

3x-4=-1      (we can then plus 4 to each side)

3x=3           (we can then divide each side by 3)


We will now do the same with equation (3):

x-4=2x+1   (we can then minus x from each side)

-4=x+1       (we can then minus 1 from each side)


We have now solved the equation x=4-|2x+1| and found that the values of x which satisfy this equation are x=1 and x=-5.

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