If the replacement set is the real numbers (R ), find the solution set of :
`8 + 3x ge 28 - 2x`.

Answer

To solve the inequality \(8 + 3x \geq 28 - 2x\), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ 8 + 3x \geq 28 - 2x \]

### Step 2: Add \(2x\) to both sides
To eliminate the \(-2x\) on the right side, we add \(2x\) to both sides:
\[ 8 + 3x + 2x \geq 28 - 2x + 2x \]
This simplifies to:
\[ 8 + 5x \geq 28 \]

### Step 3: Subtract \(8\) from both sides
Next, we subtract \(8\) from both sides to isolate the term with \(x\):
\[ 8 + 5x - 8 \geq 28 - 8 \]
This simplifies to:
\[ 5x \geq 20 \]

### Step 4: Divide both sides by \(5\)
Now, we divide both sides by \(5\) to solve for \(x\):
\[ \frac{5x}{5} \geq \frac{20}{5} \]
This simplifies to:
\[ x \geq 4 \]

### Step 5: Write the solution set
The solution set in interval notation is:
\[ [4, \infty) \]

### Summary of the Solution
The solution to the inequality \(8 + 3x \geq 28 - 2x\) is \(x \geq 4\), which means the solution set is all real numbers \(x\) such that \(x\) is greater than or equal to \(4\).

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