Solve the inequation `3x - 7 lt 5x - 3 ` If x `in` R and represent the solution on the number line

Explanation

To solve the inequation \(3x - 7 < 5x - 3\), we will follow these steps:

### Step 1: Rearranging the Inequation
Start with the original inequation:
\[ 3x - 7 < 5x - 3 \]
We want to isolate \(x\) on one side. First, we can subtract \(3x\) from both sides:
\[ -7 < 5x - 3x - 3 \]
This simplifies to:
\[ -7 < 2x - 3 \]

### Step 2: Isolate the term with \(x\)
Next, we will add \(3\) to both sides to isolate the term with \(x\):
\[ -7 + 3 < 2x \]
This simplifies to:
\[ -4 < 2x \]

### Step 3: Divide by the coefficient of \(x\)
Now, divide both sides by \(2\) to solve for \(x\):
\[ \frac{-4}{2} < x \]
This gives us:
\[ -2 < x \]
or equivalently,
\[ x > -2 \]

### Step 4: Writing the solution in interval notation
The solution can be expressed in interval notation as:
\[ x \in (-2, \infty) \]

### Step 5: Representing the solution on the number line
To represent this solution on the number line:
- Draw a number line.
- Mark the point \(-2\) on the line.
- Since \(x\) is greater than \(-2\), we will use an open circle at \(-2\) (indicating that \(-2\) is not included in the solution) and shade the region to the right towards infinity.

### Summary of the solution
The solution to the inequation \(3x - 7 < 5x - 3\) is:
\[ x > -2 \quad \text{or} \quad x \in (-2, \infty) \]

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