Solve the inequalities for real x and represent solution on number line
`(2x-1)/(3)le(3x-2)/(4)-(2-x)/(5)`

Answer

To solve the inequality \(\frac{2x-1}{3} \leq \frac{3x-2}{4} - \frac{2-x}{5}\), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ \frac{2x-1}{3} \leq \frac{3x-2}{4} - \frac{2-x}{5} \]

### Step 2: Find a common denominator
The denominators are 3, 4, and 5. The least common multiple (LCM) of these numbers is 60. We will multiply each term by 60 to eliminate the fractions:
\[ 60 \cdot \frac{2x-1}{3} \leq 60 \cdot \left(\frac{3x-2}{4} - \frac{2-x}{5}\right) \]

### Step 3: Simplify each term
Calculating each term:
- For the left side:
\[ 60 \cdot \frac{2x-1}{3} = 20(2x-1) = 40x - 20 \]
- For the right side:
\[ 60 \cdot \frac{3x-2}{4} = 15(3x-2) = 45x - 30 \]
\[ 60 \cdot \frac{2-x}{5} = 12(2-x) = 24 - 12x \]
Thus, the right side becomes:
\[ 45x - 30 - (24 - 12x) = 45x - 30 - 24 + 12x = 57x - 54 \]

### Step 4: Set up the new inequality
Now we have:
\[ 40x - 20 \leq 57x - 54 \]

### Step 5: Rearrange the inequality
Rearranging gives:
\[ 40x - 57x \leq -54 + 20 \]
\[ -17x \leq -34 \]

### Step 6: Divide by -17
When dividing by a negative number, we reverse the inequality sign:
\[ x \geq 2 \]

### Step 7: Write the solution
The solution is:
\[ x \geq 2 \]

### Step 8: Represent the solution on the number line
To represent this on a number line, we place a closed dot at 2 (indicating that 2 is included in the solution) and shade the region to the right towards infinity.

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