The solution set of the inequality `37-(3x+5)ge9x-8(x-3)` is

Options

A. `(-oo,2)`
B. `(-oo,-2)`
C. `(-oo,2]`
D. `(-oo,-2]`

Answer

To solve the inequality \( 37 - (3x + 5) \geq 9x - 8(x - 3) \), we will follow these steps:

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

Left side:
\[ 37 - (3x + 5) = 37 - 3x - 5 = 32 - 3x \]

Right side:
\[ 9x - 8(x - 3) = 9x - 8x + 24 = x + 24 \]

Now, the inequality becomes:
\[ 32 - 3x \geq x + 24 \]

### Step 2: Rearrange the inequality
Next, we will move all terms involving \(x\) to one side and constant terms to the other side.

Add \(3x\) to both sides:
\[ 32 \geq x + 3x + 24 \]
\[ 32 \geq 4x + 24 \]

Now, subtract \(24\) from both sides:
\[ 32 - 24 \geq 4x \]
\[ 8 \geq 4x \]

### Step 3: Solve for \(x\)
Now, divide both sides by \(4\):
\[ \frac{8}{4} \geq x \]
\[ 2 \geq x \]
or
\[ x \leq 2 \]

### Step 4: Write the solution set
The solution set for the inequality \(x \leq 2\) can be expressed in interval notation as:
\[ (-\infty, 2] \]

### Final Answer
Thus, the solution set of the inequality is:
\[ x \in (-\infty, 2] \]