Solve the following `|(x^2+2x+2)+(3x+7)|<|x^2+2x+2|+|3x+7|`

Answer

To solve the inequality

\[ |(x^2 + 2x + 2) + (3x + 7)| < |x^2 + 2x + 2| + |3x + 7|, \]

we will follow these steps:

### Step 1: Combine the expressions inside the absolute value on the left side.

We start by simplifying the left-hand side:

\[ |(x^2 + 2x + 2) + (3x + 7)| = |x^2 + 2x + 2 + 3x + 7| = |x^2 + 5x + 9|. \]

### Step 2: Rewrite the inequality.

Now, we can rewrite the inequality as:

\[ |x^2 + 5x + 9| < |x^2 + 2x + 2| + |3x + 7|. \]

### Step 3: Analyze the right-hand side.

Next, we need to analyze the right-hand side:

1. \( |x^2 + 2x + 2| \) can be simplified as it is already in a suitable form.
2. \( |3x + 7| \) is also in a suitable form.

### Step 4: Consider cases based on the expressions inside the absolute values.

We will consider different cases based on the signs of the expressions inside the absolute values.

#### Case 1: \( x^2 + 5x + 9 \geq 0 \)

In this case, the left side simplifies to:

\[ x^2 + 5x + 9 < |x^2 + 2x + 2| + |3x + 7|. \]

#### Case 2: \( x^2 + 5x + 9 < 0 \)

This case is not possible since \( x^2 + 5x + 9 \) is always positive (the discriminant \( 5^2 - 4 \cdot 1 \cdot 9 = 25 - 36 < 0 \)).

### Step 5: Solve the inequality from Case 1.

Since we only have Case 1, we can proceed with:

\[ x^2 + 5x + 9 < |x^2 + 2x + 2| + |3x + 7|. \]

Now, we will analyze the right-hand side based on the values of \( x \).

#### Sub-case 1.1: \( x \geq -\frac{7}{3} \)

In this case, both \( |x^2 + 2x + 2| \) and \( |3x + 7| \) will be positive:

\[ x^2 + 5x + 9 < (x^2 + 2x + 2) + (3x + 7). \]

This simplifies to:

\[ x^2 + 5x + 9 < x^2 + 5x + 9, \]

which is not possible.

#### Sub-case 1.2: \( x < -\frac{7}{3} \)

In this case, we have:

\[ x^2 + 5x + 9 < -(x^2 + 2x + 2) - (3x + 7). \]

This simplifies to:

\[ x^2 + 5x + 9 < -x^2 - 2x - 2 - 3x - 7. \]

Combining like terms gives:

\[ x^2 + 5x + 9 < -x^2 - 5x - 9. \]

Adding \( x^2 + 5x + 9 \) to both sides:

\[ 2x^2 + 10x + 18 < 0. \]

Dividing the entire inequality by 2:

\[ x^2 + 5x + 9 < 0. \]

### Step 6: Analyze the quadratic inequality.

The quadratic \( x^2 + 5x + 9 \) has no real roots (as shown earlier), thus it is always positive.

### Conclusion:

Since both cases lead to contradictions or impossible scenarios, the original inequality has no solution.