X+22<=(x+2)²

To solve the inequality x + 22 ≤ (x + 2)², we can start by expanding the squared term on the right side of the inequality:

x + 22 ≤ x² + 4x + 4

Next, let's gather all the terms on one side of the inequality:

0 ≤ x² + 4x + 4 - x - 22

Simplifying further:

0 ≤ x² + 3x - 18

Now, to solve this quadratic inequality, we can set it equal to zero and find the critical points. To do this, we can rearrange the inequality by subtracting the entire expression on the right from both sides:

x² + 3x - 18 ≤ 0

Now, let's find the critical points by factoring or using the quadratic formula. Factoring the quadratic expression, we have:

(x + 6)(x - 3) ≤ 0

To determine the sign of the expression for different intervals of x, we can use a sign chart or test points.

Test x = -7: (-7 + 6)(-7 - 3) ≤ 0, which is true (negative)

Test x = 0: (0 + 6)(0 - 3) ≤ 0, which is false (positive)

Test x = 4: (4 + 6)(4 - 3) ≤ 0, which is true (negative)

Therefore, the solution to the inequality x + 22 ≤ (x + 2)² is:

-6 ≤ x ≤ 3