Solve the inequation `5x - 2 ge 3x - 1` If x `in` R and represent the solution on the number line

Explanation

To solve the inequation \(5x - 2 \geq 3x - 1\), we will follow these steps:

### Step 1: Rewrite the Inequation
Start with the given inequation:
\[ 5x - 2 \geq 3x - 1 \]

### Step 2: Move all terms involving \(x\) to one side
Subtract \(3x\) from both sides:
\[ 5x - 3x - 2 \geq -1 \]
This simplifies to:
\[ 2x - 2 \geq -1 \]

### Step 3: Move constant terms to the other side
Add \(2\) to both sides:
\[ 2x \geq -1 + 2 \]
This simplifies to:
\[ 2x \geq 1 \]

### Step 4: Solve for \(x\)
Now, divide both sides by \(2\):
\[ x \geq \frac{1}{2} \]

### Step 5: Represent the solution on the number line
The solution \(x \geq \frac{1}{2}\) means that \(x\) can take any value starting from \(\frac{1}{2}\) and going to positive infinity. On the number line, this is represented as a closed circle at \(\frac{1}{2}\) (indicating that \(\frac{1}{2}\) is included in the solution) and a line extending to the right towards positive infinity.

### Final Answer
The solution set is:
\[ x \in \left[\frac{1}{2}, \infty\right) \]

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