Question

Find the solution to the inequality |x + 1|² +2|x+2| ≥ 2. What are the values of x that satisfy this inequality?"

1.x ≤ -3 or x ≥ -1
2.x ≤ -2 or x ≥ 0
3.x ≤ -4 or x ≥ -1
4.x ≤ -3 or x ≥ 0

Answer 1

To find the solution to the inequality |x + 1|² +2|x+2| ≥ 2, we can first simplify it by completing the square and then solve for the values of x that make the expression less than or equal to 0. The values of x that satisfy the inequality are x ≤ -1 or x ≥ 1.

Step-by-step explanation:

To find the solution to the inequality |x + 1|² +2|x+2| ≥ 2, we can first simplify it by completing the square. This condition simplifies to 2(x² − 1)² ≤ 0. We can solve this inequality by finding the values of x that make the expression on the left side less than or equal to 0.

The expression (x² − 1) will be non-negative for values of x that are greater than or equal to -1 and less than or equal to 1. So the inequality is satisfied when x is between -1 and 1. Since the expression is squared, it will be equal to 0 when x is -1 or 1. Therefore, the values of x that satisfy the inequality are x ≤ -1 or x ≥ 1.