X+22<=(x+2)²

To solve this inequality, let's first expand the right side of the equation.

(x + 2)² = x² + 4x + 4

Now let's rewrite the inequality:

x + 22 ≤ x² + 4x + 4

Next, let's subtract x and 4 from both sides of the inequality:

22 - 4 ≤ x² + 4x - x

18 ≤ x² + 3x

To simplify further, let's write the equation in standard form:

x² + 3x - 18 ≥ 0

Now we need to find the solutions to this equation:

(x + 6)(x - 3) ≥ 0

To determine the values of x that satisfy the inequality, we need to consider the sign of the factors. The inequality will be true when either both factors are positive or both factors are negative.

Case 1: (x + 6) > 0 and (x - 3) > 0 x + 6 > 0 => x > -6 x - 3 > 0 => x > 3

Since both conditions are true for x > 3, it is a valid solution.

Case 2: (x + 6) < 0 and (x - 3) < 0 x + 6 < 0 => x < -6 x - 3 < 0 => x < 3

Both conditions are true for x < -6, so it is also a valid solution.

Therefore, the solution to the inequality x + 22 ≤ (x + 2)² is x ≤ -6 or x ≥ 3.