Questions: Do NOT convert the answer to a decimal. (x-2)/12 + 3/4 = (x+3)/3 Answer: x=

Transcript text: Do NOT convert the answer to a decimal. \[ \frac{x-2}{12}+\frac{3}{4}=\frac{x+3}{3} \] Answer: $x=$ $\square$

Solution

Solution Steps

To solve the given equation, we need to find a common denominator for the fractions involved. Once we have a common denominator, we can combine the fractions on the left side of the equation. After that, we can equate the numerators and solve for $ x $.

Step 1: Identify the Common Denominator

To solve the equation $\frac{x-2}{12} + \frac{3}{4} = \frac{x+3}{3}$, we first identify a common denominator for the fractions. The denominators are 12, 4, and 3. The least common multiple of these numbers is 12.

Step 2: Rewrite the Equation with a Common Denominator

Rewrite each term with the common denominator of 12: \[ \frac{x-2}{12} + \frac{3 \times 3}{4 \times 3} = \frac{x \times 4 + 3 \times 4}{3 \times 4} \] This simplifies to: \[ \frac{x-2}{12} + \frac{9}{12} = \frac{4x + 12}{12} \]

Step 3: Combine the Fractions on the Left Side

Combine the fractions on the left side: \[ \frac{x-2 + 9}{12} = \frac{4x + 12}{12} \] This simplifies to: \[ \frac{x + 7}{12} = \frac{4x + 12}{12} \]

Step 4: Equate the Numerators

Since the denominators are the same, equate the numerators: \[ x + 7 = 4x + 12 \]

Step 5: Solve for $ x $

Rearrange the equation to solve for $ x $: \[ x + 7 = 4x + 12 \implies 7 - 12 = 4x - x \implies -5 = 3x \implies x = -\frac{5}{3} \]