Solve: `(2x-3)/(4)+8ge2+(4x)/(3)` and show the solution set on the number line.

Answer

To solve the inequality \(\frac{2x-3}{4} + 8 \geq 2 + \frac{4x}{3}\), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ \frac{2x-3}{4} + 8 \geq 2 + \frac{4x}{3} \]

### Step 2: Eliminate the fractions
To eliminate the fractions, we can find a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Multiply every term by 12:
\[ 12 \left(\frac{2x-3}{4}\right) + 12(8) \geq 12(2) + 12\left(\frac{4x}{3}\right) \]
This simplifies to:
\[ 3(2x-3) + 96 \geq 24 + 4(4x) \]

### Step 3: Distribute and simplify
Distributing the terms gives:
\[ 6x - 9 + 96 \geq 24 + 16x \]
Combine like terms:
\[ 6x + 87 \geq 24 + 16x \]

### Step 4: Rearrange the inequality
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 6x - 16x + 87 \geq 24 \]
This simplifies to:
\[ -10x + 87 \geq 24 \]

### Step 5: Isolate \(x\)
Subtract 87 from both sides:
\[ -10x \geq 24 - 87 \]
This simplifies to:
\[ -10x \geq -63 \]
Now, divide both sides by -10. Remember, when dividing by a negative number, we must reverse the inequality sign:
\[ x \leq \frac{63}{10} \]
This simplifies to:
\[ x \leq 6.3 \]

### Step 6: Write the solution set
The solution set in interval notation is:
\[ (-\infty, 6.3] \]

### Step 7: Plot the solution on the number line
To represent this on a number line, we will shade all values to the left of 6.3, including 6.3 itself.

### Summary of the solution
The solution to the inequality \(\frac{2x-3}{4} + 8 \geq 2 + \frac{4x}{3}\) is:
\[ x \leq 6.3 \]
The solution set is represented as \((- \infty, 6.3]\) on the number line.

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