To solve this inequality, we first need to simplify both sides.
Starting with the left side:
| -1 - x^2 |
Since the absolute value of a number will always be non-negative, we can remove the absolute value signs when squared:
= (-1 - x^2)^2
= 1 + 2x^2 + x^4
Now, simplifying the right side:
| 3x - x^2 - 4 |
= |-x^2 + 3x - 4|
= (x^2 - 3x + 4)
Now we have:
1 + 2x^2 + x^4 <= x^2 - 3x + 4
Rearranging and simplifying the inequality:
x^4 + x^2 - x - 3 <= 0
Factoring the left side:
(x^2 - 2)(x^2 + 1) <= 0
The solutions for x are the values that make the inequality true. To find these solutions, we need to consider when the expression is less than or equal to zero:
(x^2 - 2)(x^2 + 1) = 0
x^2 = 2 or x^2 = -1
However, x^2 = -1 has no real solutions, so we consider x^2 = 2:
x = ±√2
Therefore, the solution to the inequality is x ≤ -√2 or x ≥ √2.
To solve this inequality, we first need to simplify both sides.
Starting with the left side:
| -1 - x^2 |
Since the absolute value of a number will always be non-negative, we can remove the absolute value signs when squared:
= (-1 - x^2)^2
= 1 + 2x^2 + x^4
Now, simplifying the right side:
| 3x - x^2 - 4 |
= |-x^2 + 3x - 4|
= (x^2 - 3x + 4)
Now we have:
1 + 2x^2 + x^4 <= x^2 - 3x + 4
Rearranging and simplifying the inequality:
x^4 + x^2 - x - 3 <= 0
Factoring the left side:
(x^2 - 2)(x^2 + 1) <= 0
The solutions for x are the values that make the inequality true. To find these solutions, we need to consider when the expression is less than or equal to zero:
(x^2 - 2)(x^2 + 1) = 0
x^2 = 2 or x^2 = -1
However, x^2 = -1 has no real solutions, so we consider x^2 = 2:
x = ±√2
Therefore, the solution to the inequality is x ≤ -√2 or x ≥ √2.