2.3 $$ \frac{2(2-x)}{x-2}=\frac{x+3}{2} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{2(2 - x)}{x - 2} = \frac{x + 3}{2}
$$

let's follow these steps:

### Step 1: Simplify the Left Side

Notice that $$2(2 - x)$$ can be rewritten as $$2(-x + 2) = -2(x - 2)$$. So, the equation becomes:

$$
\frac{-2(x - 2)}{x - 2} = \frac{x + 3}{2}
$$

Assuming $$x \neq 2$$ (to avoid division by zero), the $$(x - 2)$$ terms cancel out:

$$
-2 = \frac{x + 3}{2}
$$

### Step 2: Solve for $$x$$

Multiply both sides of the equation by 2 to eliminate the fraction:

$$
-2 \times 2 = x + 3
$$
$$
-4 = x + 3
$$

Subtract 3 from both sides:

$$
-4 - 3 = x
$$
$$
x = -7
$$

### Step 3: Verify the Solution

It's always good practice to check if the solution doesn't make any denominator zero in the original equation.

- Original denominator: $$x - 2$$
- Substitute $$x = -7$$:
  $$
  -7 - 2 = -9 \neq 0
  $$

Since the denominator is not zero, $$x = -7$$ is a valid solution.

### Final Answer

The solution to the equation is:

$$
x = -7
$$
$$ x = -7 $$

2.3