Solve : `(5x)/(4)-1 le (4x-1)/(3)`

Explanation

To solve the inequality \(\frac{5x}{4} - 1 \leq \frac{4x - 1}{3}\), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ \frac{5x}{4} - 1 \leq \frac{4x - 1}{3} \]

### Step 2: Move all terms to one side
We can rearrange the inequality by moving all terms to one side:
\[ \frac{5x}{4} - \frac{4x - 1}{3} - 1 \leq 0 \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 4 and 3 is 12. We will rewrite each term with a denominator of 12:
\[ \frac{5x}{4} = \frac{15x}{12}, \quad \frac{4x - 1}{3} = \frac{4(4x - 1)}{12} = \frac{16x - 4}{12} \]
Thus, the inequality becomes:
\[ \frac{15x}{12} - \frac{16x - 4}{12} - 1 \leq 0 \]

### Step 4: Combine the fractions
Now, combine the fractions:
\[ \frac{15x - (16x - 4)}{12} - 1 \leq 0 \]
This simplifies to:
\[ \frac{15x - 16x + 4}{12} - 1 \leq 0 \]
Which further simplifies to:
\[ \frac{-x + 4}{12} - 1 \leq 0 \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply the entire inequality by 12 (note that 12 is positive, so the inequality sign remains the same):
\[ -x + 4 - 12 \leq 0 \]
This simplifies to:
\[ -x - 8 \leq 0 \]

### Step 6: Solve for \(x\)
Rearranging gives:
\[ -x \leq 8 \]
Multiplying by -1 (which reverses the inequality):
\[ x \geq -8 \]

### Step 7: Write the solution in interval notation
The solution in interval notation is:
\[ x \in [-8, \infty) \]

### Final Answer
The final solution is:
\[ x \geq -8 \quad \text{or} \quad x \in [-8, \infty) \]

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