solve the inequation `(2x-3)/(4)+9ge3+(4x)/(3)` and represent this solution on number line .

Explanation

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

### Step 1: Rearrange the Inequality
Start by moving all terms involving \(x\) to one side and constant terms to the other side.

\[ \frac{2x - 3}{4} + 9 - \frac{4x}{3} \geq 3 \]

Subtract \(3\) from both sides:

\[ \frac{2x - 3}{4} + 9 - 3 \geq \frac{4x}{3} \]

This simplifies to:

\[ \frac{2x - 3}{4} + 6 \geq \frac{4x}{3} \]

### Step 2: Combine Like Terms
Now, we can isolate the \(x\) terms. First, we can subtract \(6\) from both sides:

\[ \frac{2x - 3}{4} \geq \frac{4x}{3} - 6 \]

### Step 3: Find a Common Denominator
To eliminate the fractions, we need a common denominator. The least common multiple of \(4\) and \(3\) is \(12\). Multiply every term by \(12\):

\[ 12 \cdot \frac{2x - 3}{4} \geq 12 \cdot \left(\frac{4x}{3} - 6\right) \]

This simplifies to:

\[ 3(2x - 3) \geq 4(4x) - 72 \]

### Step 4: Distribute and Simplify
Distributing both sides gives:

\[ 6x - 9 \geq 16x - 72 \]

### Step 5: Move All \(x\) Terms to One Side
Now, we can move all \(x\) terms to one side by subtracting \(16x\) from both sides:

\[ 6x - 16x - 9 \geq -72 \]

This simplifies to:

\[ -10x - 9 \geq -72 \]

### Step 6: Isolate \(x\)
Next, add \(9\) to both sides:

\[ -10x \geq -72 + 9 \]

This simplifies to:

\[ -10x \geq -63 \]

Now, divide both sides by \(-10\). Remember, when dividing by a negative number, we reverse the inequality:

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

### Step 7: Final Result
Thus, we have:

\[ x \leq 6.3 \]

In interval notation, this is represented as:

\[ x \in (-\infty, 6.3] \]

### Step 8: Represent on a Number Line
To represent this solution on a number line, we draw a line with an arrow pointing to the left from \(6.3\), indicating all values less than or equal to \(6.3\). We place a closed dot on \(6.3\) to indicate that it is included in the solution set.

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