Solve the inequalities for real x : `4x + 3 < 6x + 7`

Explanation

To solve the inequality \( 4x + 3 < 6x + 7 \), we will follow these steps:

### Step 1: Rearranging the inequality
Start by moving all terms involving \( x \) to one side and constant terms to the other side.

\[ 4x + 3 < 6x + 7 \]

Subtract \( 6x \) from both sides:

\[ 4x - 6x + 3 < 7 \]

This simplifies to:

\[ -2x + 3 < 7 \]

### Step 2: Isolate the term with \( x \)
Next, subtract \( 3 \) from both sides to isolate the term with \( x \):

\[ -2x < 7 - 3 \]

This simplifies to:

\[ -2x < 4 \]

### Step 3: Divide by the coefficient of \( x \)
Now, divide both sides by \( -2 \). Remember that when you divide by a negative number, you must reverse the inequality sign:

\[ x > \frac{4}{-2} \]

This simplifies to:

\[ x > -2 \]

### Step 4: Write the solution in interval notation
The solution \( x > -2 \) can be expressed in interval notation as:

\[ x \in (-2, \infty) \]

### Final Answer
The solution to the inequality \( 4x + 3 < 6x + 7 \) is:

\[ x > -2 \quad \text{or} \quad x \in (-2, \infty) \]

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