Question

simplify the expression.
\frac{4}{x + 3} + \frac{4}{3} + 6
step 1
multiply each term by 3(x + 3). also divide the entire expression by 3(x + 3).
\frac{4}{x + 3} \cdot \square + \frac{4}{3} \cdot \square + 6 \cdot \square
simplify each term in the numerator.
\frac{\square + \square (x + 3) + \square (x + 3)}{3(x + 3)}

Explanation:

Step1: Multiply terms by LCD

To eliminate denominators, multiply each term by the least common denominator $3(x+3)$, and write the entire expression over $3(x+3)$:
$$\frac{\frac{4}{x+3} \cdot 3(x+3) + \frac{4}{3} \cdot 3(x+3) + 6 \cdot 3(x+3)}{3(x+3)}$$

Step2: Fill in the blanks

Identify the multiplier for each term:

• For $\frac{4}{x+3}$: multiplier is $3(x+3)$
• For $\frac{4}{3}$: multiplier is $3(x+3)$
• For $6$: multiplier is $3(x+3)$

The intermediate numerator form before simplifying is:
$$\frac{4 \cdot 3 + 4 \cdot (x+3) + 6 \cdot 3(x+3)}{3(x+3)}$$

Step3: Simplify numerator terms

Calculate each term in the numerator:

• $4 \cdot 3 = 12$
• $4 \cdot (x+3) = 4x + 12$
• $6 \cdot 3(x+3) = 18(x+3) = 18x + 54$

Step4: Combine numerator terms

Add the simplified numerator terms:
$$12 + 4x + 12 + 18x + 54 = (4x+18x) + (12+12+54) = 22x + 78$$

Step5: Final simplification

Factor the numerator and cancel common factors (if possible):
$$\frac{22x + 78}{3(x+3)} = \frac{2(11x + 39)}{3(x+3)}$$

Answer:

Filled blanks (Step1):
$\frac{4}{x+3} \cdot \boldsymbol{3(x+3)} + \frac{4}{3} \cdot \boldsymbol{3(x+3)} + 6 \cdot \boldsymbol{3(x+3)}$
$\boldsymbol{\overline{3(x+3)}}$

Filled blanks (Simplify numerator):
$\frac{\boldsymbol{12} + \boldsymbol{4}(x+3) + \boldsymbol{18}(x+3)}{3(x+3)}$

Final simplified expression: $\frac{2(11x + 39)}{3(x+3)}$