The set of points (x, y) which satisfy |x| 1, |y| 1, and |y| = |x|

From the question stem it is clear that x and y are either -1, 0 or a positive / negative decimal value.

Now let's substitutte some values for x and y.

Let x = -1

Then |y| = |-1| + -1
|y| = 0

Thus (x,y) = (-1,0) which has a length of 1 when plotted on the number line.

Now, let |x| = x

Then |y| = |0.5| + 0.5
|y| = 1

Now the actual value of y in the modulus can be either 1 our -1, but the value of x has to be non negative because of the constraint placed by the modulus (x is on the RHS side of the equation and its addition gives a non negative value).

Thus, length of (x,y) = (1, 0.5), which when we use the distance formula is \(\sqrt{(0.5)^2 + (1)^2} \)
i.e, \(\sqrt{5/4}\)
Which can also be written as \(\frac{\sqrt{5}}{ 2}\) (since 4's square root is 2).

Since there are two such values, i.e, (0.5, 1) and (0.5, -1)

\(\frac{\sqrt{5} }{ 2}\) * 2 = \(\sqrt{5}\)

Thus, adding up all the lengths we get 1 + \(\sqrt{5}\)