|x + 2| + |2 - x| < 8
To solve the inequality |x + 2| + |2 - x| < 8, we'll need to consider different cases based on the value of x.
1. Case 1: x ≥ 2
When x is greater than or equal to 2, |x + 2| simplifies to x + 2, and |2 - x| simplifies to 2 - x. Therefore, the inequality becomes:
(x + 2) + (2 - x) < 8
Simplifying further:
x + 2 + 2 - x < 8
4 < 8
In this case, the inequality is always true since 4 is always less than 8. So any value of x greater than or equal to 2 satisfies the inequality.
2. Case 2: -2 ≤ x < 2
When x is between -2 and 2 (inclusive), |x + 2| simplifies to x + 2, and |2 - x| simplifies to 2 - x as well. Therefore, the inequality becomes:
(x + 2) + (2 - x) < 8
Simplifying further:
x + 2 + 2 - x < 8
4 < 8
Similar to Case 1, the inequality is true for any value of x between -2 and 2.
3. Case 3: x < -2
When x is less than -2, |x + 2| simplifies to -(x + 2), and |2 - x| simplifies to 2 - x. Therefore, the inequality becomes:
-(x + 2) + (2 - x) < 8
Simplifying further:
-x - 2 + 2 - x < 8
-2x < 8
x > -4
In this case, the inequality is true for any value of x greater than -4.
Combining the results of all three cases, we have:
x ≥ 2 or -2 ≤ x < 2 or x > -4
So, the solution to the inequality |x + 2| + |2 - x| < 8 is x ≥ 2 or -2 ≤ x < 2 or x > -4.