Solve the following equation. $$ \frac{1}{2}(x-3)=\frac{1}{3}(x+4) $$ $$ x=\square $$ (Type an integer or a simplified fraction.)

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{1}{2}(x - 3) = \frac{1}{3}(x + 4)
$$

**Step 1: Eliminate the Fractions**

Find the least common multiple (LCM) of the denominators 2 and 3, which is 6. Multiply both sides of the equation by 6 to eliminate the fractions:

$$
6 \times \frac{1}{2}(x - 3) = 6 \times \frac{1}{3}(x + 4)
$$

**Step 2: Simplify Both Sides**

$$
3(x - 3) = 2(x + 4)
$$

**Step 3: Distribute the Multiplication**

$$
3x - 9 = 2x + 8
$$

**Step 4: Solve for $$ x $$**

Subtract $$ 2x $$ from both sides:

$$
3x - 2x - 9 = 8
$$
$$
x - 9 = 8
$$

Add 9 to both sides:

$$
x = 17
$$

**Verification:**

Plugging $$ x = 17 $$ back into the original equation:

$$
\frac{1}{2}(17 - 3) = \frac{1}{3}(17 + 4)
$$
$$
\frac{1}{2}(14) = \frac{1}{3}(21)
$$
$$
7 = 7
$$

Both sides are equal, so $$ x = 17 $$ is the correct solution.

$$
x = 17
$$
$$ x = 17 $$

Solve the following equation. \[ \frac{1}{2}(x-3)=\frac{1}{3}(x+4) \] \( x=\square \) (Type an integer or a simplified fraction.)