To solve this inequality, we first need to rewrite it in a more simplified form. The given inequality is:

lg^2(x) - lg(x) - 6 ≥ 0

Let's substitute lg(x) as t:

t = lg(x)

Now, the inequality becomes:

t^2 - t - 6 ≥ 0

We can rewrite this quadratic inequality in factored form:

(t - 3)(t + 2) ≥ 0

Now, we need to find the critical points by setting each factor to zero:

t - 3 = 0
t = 3

t + 2 = 0
t = -2

So, the critical points are t = 3 and t = -2. We can now create intervals on the number line to test the sign of the expression:

-∞ -2 3 +∞

From the number line, we can see that the inequality is true for t ≤ -2 and t ≥ 3. Now, let's substitute back t as lg(x):

lg(x) ≤ -2 and lg(x) ≥ 3

Solving for x:

x ≤ 10^(-2) and x ≥ 10^3

x ≤ 0.01 and x ≥ 1000

Therefore, the solution for the inequality lg^2(x) - lg(x) - 6 ≥ 0 is:

x ≤ 0.01 or x ≥ 1000.