Solve the inequalities for real x : `3x - 7 > 5x - 1`

Explanation

To solve the inequality \(3x - 7 > 5x - 1\), we will follow these steps:

### Step 1: Rearrange the inequality
We start with the given inequality:
\[ 3x - 7 > 5x - 1 \]
To isolate \(x\), we will move all terms involving \(x\) to one side and constant terms to the other side. We can do this by subtracting \(5x\) from both sides:
\[ 3x - 5x - 7 > -1 \]

### Step 2: Simplify the inequality
Now, simplify the left side:
\[ -2x - 7 > -1 \]
Next, we will add \(7\) to both sides to isolate the term with \(x\):
\[ -2x > -1 + 7 \]

### Step 3: Simplify further
Now, simplify the right side:
\[ -2x > 6 \]

### Step 4: Divide by -2
To solve for \(x\), we will divide both sides by \(-2\). Remember that when we divide by a negative number, we must reverse the inequality sign:
\[ x < \frac{6}{-2} \]

### Step 5: Simplify the division
Now, simplify the right side:
\[ x < -3 \]

### Step 6: Write the solution in interval notation
The solution to the inequality is that \(x\) can take any value less than \(-3\). In interval notation, this is expressed as:
\[ x \in (-\infty, -3) \]

### Final Answer
Thus, the solution to the inequality \(3x - 7 > 5x - 1\) is:
\[ x \in (-\infty, -3) \]

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