(x-3)^2 >=(x+6)*(x-1)

To solve the inequality (x-3)^2 ≥ (x+6)*(x-1), you need to expand the expressions and then bring all terms to one side to form a quadratic inequality.

First, we will expand both sides of the inequality:

(x-3)^2 becomes x^2 - 6x + 9 (x+6)*(x-1) becomes x^2 + 6x - 6x - 6, which simplifies to x^2 - 6

Now, let's rewrite the inequality:

x^2 - 6x + 9 ≥ x^2 - 6

Next, we'll subtract x^2 from both sides:

• 6x + 9 ≥ -6

Now, add 6 to both sides:

• 6x + 9 + 6 ≥ 0

This simplifies to:

-6x + 15 ≥ 0

Divide by -6 and reverse the inequality sign because we're dividing by a negative number:

x - 15/6 ≤ 0

Further simplification gives us:

x ≤ 15/6

x ≤ 5/2 or 2.5

Therefore, the solution to the inequality (x-3)^2 ≥ (x+6)*(x-1) is x ≤ 2.5.

To check this solution and to better understand the inequality, you may consider creating a number line and testing values from each interval to ensure that they satisfy the original inequality. Alternatively, for a more exact solution, you could use a graphing approach to see where the parabola (x-3)^2 intersects and is above the line (x+6)(x-1). This will give you a visual confirmation of the solution range.