Solve the inequalities for real x and represent solution on number line
`(2x-3)/(4)+9ge3+(4x)/(3)`,`xinR`

Answer

To solve the inequality \(\frac{2x-3}{4} + 9 \geq 3 + \frac{4x}{3}\), we will follow these steps:

### Step 1: Clear the fractions
To eliminate the fractions, we can find a common denominator. The least common multiple (LCM) of 4 and 3 is 12. We will multiply each term in the inequality by 12.

\[ 12 \left(\frac{2x-3}{4}\right) + 12(9) \geq 12(3) + 12\left(\frac{4x}{3}\right) \]

### Step 2: Simplify the equation
Now, we simplify each term:

\[ 3(2x - 3) + 108 \geq 36 + 4(4x) \]

This leads to:

\[ 6x - 9 + 108 \geq 36 + 16x \]

### Step 3: Combine like terms
Combine the constants on the left side:

\[ 6x + 99 \geq 36 + 16x \]

### Step 4: Rearrange the inequality
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:

\[ 99 - 36 \geq 16x - 6x \]

This simplifies to:

\[ 63 \geq 10x \]

### Step 5: Solve for \(x\)
Now, divide both sides by 10:

\[ \frac{63}{10} \geq x \]

This can also be written as:

\[ x \leq 6.3 \]

### Step 6: Represent the solution on a number line
To represent \(x \leq 6.3\) on a number line, we will draw a closed circle at 6.3 (indicating that 6.3 is included in the solution) and shade everything to the left of 6.3.

### Final Answer
The solution to the inequality is:

\[ x \leq 6.3 \]