Explanation:
To solve the inequality |2x + 5| < 2 - x, we need to split it into two cases:
When 2x + 5 > 0:
In this case, the absolute value is positive, so we can simplify the inequality as follows:
2x + 5 < 2 - x
Expanding the left side:
3x + 5 < 2
Subtracting 5 from both sides:
3x < -3
Dividing both sides by 3:
x < -1
So the solution for this case is x < -1.
When 2x + 5 < 0:
In this case, the absolute value is negative, so we can simplify the inequality as follows:
(2x + 5) < 2 - x
Expanding the left side:
-2x - 5 < 2 - x
Adding x to both sides:
-2x - 5 + x < 2
Combining like terms:
-x - 5 < 2
Adding 5 to both sides:
-x < 7
Dividing both sides by -1:
x > -7
So the solution for this case is x > -7.
Taking the union of the two solutions, we get:
-7 < x < -1.
This means that x is between -7 and -1, excluding -7 and -1.