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

Answer

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

### Step 1: Simplify both sides of the inequality
Start by simplifying the left-hand side and the right-hand side.

**Left-hand side:**
\[ \frac{2x + 3}{4} - 3 = \frac{2x + 3}{4} - \frac{12}{4} = \frac{2x + 3 - 12}{4} = \frac{2x - 9}{4} \]

**Right-hand side:**
\[ \frac{x - 4}{3} - 2 = \frac{x - 4}{3} - \frac{6}{3} = \frac{x - 4 - 6}{3} = \frac{x - 10}{3} \]

So, the inequality now looks like:
\[ \frac{2x - 9}{4} < \frac{x - 10}{3} \]

### Step 2: Eliminate the fractions
To eliminate the fractions, we can find a common multiple of the denominators (4 and 3), which is 12. Multiply both sides of the inequality by 12:
\[ 12 \cdot \frac{2x - 9}{4} < 12 \cdot \frac{x - 10}{3} \]

This simplifies to:
\[ 3(2x - 9) < 4(x - 10) \]

### Step 3: Distribute and simplify
Distributing both sides gives:
\[ 6x - 27 < 4x - 40 \]

### Step 4: Rearranging the inequality
Now, we will rearrange the inequality to isolate \(x\):
\[ 6x - 4x < -40 + 27 \]
\[ 2x < -13 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 2:
\[ x < -\frac{13}{2} \]
\[ x < -6.5 \]

### Step 6: Represent the solution on the number line
The solution \(x < -6.5\) means that \(x\) can take any value less than \(-6.5\). On the number line, we represent this by shading the area to the left of \(-6.5\) and placing an open circle at \(-6.5\) to indicate that it is not included in the solution.

### Final Solution
The solution to the inequality is:
\[ x \in (-\infty, -6.5) \]

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