Solve : ` (5x)/(4)gt (4x-1)/(3) and 4x + 3 lt (2x + 1)/(3)`

Explanation

To solve the inequalities \( \frac{5x}{4} > \frac{4x - 1}{3} \) and \( 4x + 3 < \frac{2x + 1}{3} \), we will tackle each inequality step by step.

### Step 1: Solve the first inequality \( \frac{5x}{4} > \frac{4x - 1}{3} \)

1. **Multiply both sides by 12** (the least common multiple of 4 and 3) to eliminate the fractions:
\[ 12 \cdot \frac{5x}{4} > 12 \cdot \frac{4x - 1}{3} \]

2. **Simplify**:
\[ 3 \cdot 5x > 4(4x - 1) \]
\[ 15x > 16x - 4 \]

3. **Rearrange the inequality** by subtracting \( 16x \) from both sides:
\[ 15x - 16x > -4 \]
\[ -x > -4 \]

4. **Multiply by -1** (remember to reverse the inequality sign):
\[ x < 4 \]

### Step 2: Solve the second inequality \( 4x + 3 < \frac{2x + 1}{3} \)

1. **Multiply both sides by 3** to eliminate the fraction:
\[ 3(4x + 3) < 2x + 1 \]

2. **Simplify**:
\[ 12x + 9 < 2x + 1 \]

3. **Rearrange the inequality** by subtracting \( 2x \) from both sides:
\[ 12x - 2x + 9 < 1 \]
\[ 10x + 9 < 1 \]

4. **Subtract 9 from both sides**:
\[ 10x < 1 - 9 \]
\[ 10x < -8 \]

5. **Divide by 10**:
\[ x < -0.8 \]

### Step 3: Combine the results

We have two inequalities:
1. \( x < 4 \)
2. \( x < -0.8 \)

To satisfy both inequalities, we take the intersection:
\[ x < -0.8 \]

### Final Answer
The solution is:
\[ x \in (-\infty, -0.8) \]