Solve the inequalities for real x and represent solution on number line
`(2x+3)/(x-3)le4`

Answer

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

### Step 1: Rewrite the inequality
Start by rewriting the inequality:
\[ \frac{2x + 3}{x - 3} \leq 4 \]

### Step 2: Move all terms to one side
Multiply both sides by \(x - 3\) (noting that we need to consider the sign of \(x - 3\) later):
\[ 2x + 3 \leq 4(x - 3) \]

### Step 3: Expand the right side
Now, expand the right side:
\[ 2x + 3 \leq 4x - 12 \]

### Step 4: Rearrange the inequality
Rearranging gives:
\[ 2x + 3 - 4x + 12 \leq 0 \]
which simplifies to:
\[ -2x + 15 \leq 0 \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):
\[ 15 \leq 2x \]
or
\[ x \geq \frac{15}{2} \]
which simplifies to:
\[ x \geq 7.5 \]

### Step 6: Consider the domain
Since we multiplied by \(x - 3\), we need to ensure that \(x - 3 \neq 0\), which means \(x \neq 3\). However, since \(7.5 > 3\), this condition is satisfied.

### Step 7: Write the final solution
The solution to the inequality is:
\[ x \geq 7.5 \]

### Step 8: Represent on the number line
On the number line, we represent this solution by shading to the right of \(7.5\) and including \(7.5\) itself (indicated by a closed dot).

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