Solve the following inequalities and show the graph of the solution set on number line:
(i) `3x-2lt2x+1`
(ii) `5x-3ge3x-5`.

Answer

Let's solve the given inequalities step by step.

### (i) Solve the inequality: \( 3x - 2 < 2x + 1 \)

**Step 1:** Start with the original inequality:
\[ 3x - 2 < 2x + 1 \]

**Step 2:** Subtract \(2x\) from both sides:
\[ 3x - 2x - 2 < 1 \]
This simplifies to:
\[ x - 2 < 1 \]

**Step 3:** Add 2 to both sides:
\[ x < 1 + 2 \]
This simplifies to:
\[ x < 3 \]

### Graphing the solution on the number line:
- The solution \(x < 3\) means that all values less than 3 are included in the solution set.
- We represent this on the number line with an open circle at 3 (indicating that 3 is not included) and shading to the left.

### (ii) Solve the inequality: \( 5x - 3 \geq 3x - 5 \)

**Step 1:** Start with the original inequality:
\[ 5x - 3 \geq 3x - 5 \]

**Step 2:** Subtract \(3x\) from both sides:
\[ 5x - 3x - 3 \geq -5 \]
This simplifies to:
\[ 2x - 3 \geq -5 \]

**Step 3:** Add 3 to both sides:
\[ 2x \geq -5 + 3 \]
This simplifies to:
\[ 2x \geq -2 \]

**Step 4:** Divide both sides by 2:
\[ x \geq -1 \]

### Graphing the solution on the number line:
- The solution \(x \geq -1\) means that all values greater than or equal to -1 are included in the solution set.
- We represent this on the number line with a closed circle at -1 (indicating that -1 is included) and shading to the right.

### Summary of Solutions:
1. For \(3x - 2 < 2x + 1\), the solution is \(x < 3\).
2. For \(5x - 3 \geq 3x - 5\), the solution is \(x \geq -1\).