Question

expanding expressions continued
11 (1/3)(9x + 16 + 2) + 2x 7x + 14 - 2(x + 4) x - 3 + 7(x + 3) - 3x - 12
12 use two different methods to expand (1/4)(x + 2x + 16 - 8).

Explanation:

For the expression \(\frac{1}{3}(9x + 16 + 2)+2x\)

Step1: Simplify inside the parentheses

First, combine the constant terms inside the parentheses: \(16 + 2 = 18\), so the expression becomes \(\frac{1}{3}(9x + 18)+2x\).

Step2: Distribute the \(\frac{1}{3}\)

Using the distributive property \(a(b + c)=ab+ac\), we have \(\frac{1}{3}\times9x+\frac{1}{3}\times18 + 2x\). Calculating each term: \(\frac{1}{3}\times9x = 3x\) and \(\frac{1}{3}\times18=6\), so now the expression is \(3x + 6+2x\).

Step3: Combine like terms

Combine the \(x\)-terms: \(3x+2x = 5x\), so the simplified expression is \(5x + 6\).

For the expression \(7x + 14-2(x + 4)\)

Step1: Distribute the \(-2\)

Using the distributive property, \(-2(x + 4)=-2x-8\), so the expression becomes \(7x + 14-2x - 8\).

Step2: Combine like terms

Combine the \(x\)-terms: \(7x-2x = 5x\), and combine the constant terms: \(14 - 8=6\). So the simplified expression is \(5x+6\).

For the expression \(x - 3+7(x + 3)-3x - 12\)

Step1: Distribute the \(7\)

Using the distributive property, \(7(x + 3)=7x+21\), so the expression becomes \(x - 3+7x + 21-3x - 12\).

Step2: Combine like terms

Combine the \(x\)-terms: \(x+7x-3x=(1 + 7-3)x = 5x\). Combine the constant terms: \(-3 + 21-12=(-3-12)+21=-15 + 21 = 6\). So the simplified expression is \(5x + 6\).

For the expression \(\frac{1}{4}(x + 2x+16 - 8)\) (using two methods)

Method 1: Combine like terms first, then distribute

Answer:

Step1: Distribute the \(\frac{1}{4}\)

Using the distributive property, \(\frac{1}{4}(x + 2x+16 - 8)=\frac{1}{4}x+\frac{1}{4}(2x)+\frac{1}{4}\times16-\frac{1}{4}\times8\).

Step2: Simplify each term

\(\frac{1}{4}x+\frac{2}{4}x + 4-2\). Simplify the \(x\)-terms: \(\frac{1 + 2}{4}x=\frac{3}{4}x\), and simplify the constant terms: \(4-2 = 2\). So the expression is \(\frac{3}{4}x+2\).