Sure, let's break down each expression step by step to simplify them:
### Part a: [tex]\(4(x-6) - 30\)[/tex]
1. Distribute the 4 into the parentheses:
[tex]\[ 4 \cdot x - 4 \cdot 6 - 30 \][/tex]
2. Simplify the multiplication:
[tex]\[ 4x - 24 - 30 \][/tex]
3. Combine the constants (-24 and -30):
[tex]\[ 4x - 54 \][/tex]
So, the simplified expression is:
[tex]\[ 4(x-6) - 30 = 4x - 54 \][/tex]
### Part b: [tex]\(\frac{1}{4}y + \frac{3}{4}(y-8)\)[/tex]
1. Distribute [tex]\(\frac{3}{4}\)[/tex] into the parentheses:
[tex]\[ \frac{1}{4}y + \frac{3}{4}y - \frac{3}{4} \cdot 8 \][/tex]
2. Simplify the multiplication:
[tex]\[ \frac{1}{4}y + \frac{3}{4}y - 6 \][/tex]
3. Combine the like terms ([tex]\(\frac{1}{4}y\)[/tex] and [tex]\(\frac{3}{4}y\)[/tex]):
[tex]\[ \left(\frac{1}{4} + \frac{3}{4}\right)y - 6 \][/tex]
[tex]\[ y - 6 \][/tex]
So, the simplified expression is:
[tex]\[ \frac{1}{4}y + \frac{3}{4}(y-8) = y - 6 \][/tex]
### Part c: [tex]\(-4 + 7y - 3y - 5\)[/tex]
1. Combine the like terms (7y and -3y):
[tex]\[ -4 + 4y - 5 \][/tex]
2. Combine the constants (-4 and -5):
[tex]\[ 4y - 9 \][/tex]
So, the simplified expression is:
[tex]\[ -4 + 7y - 3y - 5 = 4y - 9 \][/tex]
### Part d: [tex]\(9y - 4x + 3y + 4x\)[/tex]
1. Combine the like terms (9y and 3y):
[tex]\[ (9 + 3)y - 4x + 4x \][/tex]
[tex]\[ 12y - 4x + 4x \][/tex]
2. Combine the like terms (-4x and 4x):
[tex]\[ 12y + 0x \][/tex]
[tex]\[ 12y \][/tex]
So, the simplified expression is:
[tex]\[ 9y - 4x + 3y + 4x = 12y \][/tex]
### Part e: [tex]\(\frac{3}{2}x - \frac{1}{2}(x + 4)\)[/tex]
1. Distribute [tex]\(\frac{1}{2}\)[/tex] into the parentheses:
[tex]\[ \frac{3}{2}x - \left(\frac{1}{2}x + \frac{1}{2} \cdot 4\right) \][/tex]
2. Simplify the multiplication:
[tex]\[ \frac{3}{2}x - \frac{1}{2}x - 2 \][/tex]
3. Combine the like terms ([tex]\(\frac{3}{2}x\)[/tex] and [tex]\(\frac{1}{2}x\)[/tex]):
[tex]\[ \left(\frac{3}{2} - \frac{1}{2}\right)x - 2 \][/tex]
[tex]\[ \frac{2}{2}x - 2 \][/tex]
[tex]\[ x - 2 \][/tex]
So, the simplified expression is:
[tex]\[ \frac{3}{2}x - \frac{1}{2}(x + 4) = x - 2 \][/tex]
### Part f: [tex]\(-8(x + 3)\)[/tex]
1. Distribute -8 into the parentheses:
[tex]\[ -8 \cdot x - 8 \cdot 3 \][/tex]
2. Simplify the multiplication:
[tex]\[ -8x - 24 \][/tex]
So, the simplified expression is:
[tex]\[ -8(x + 3) = -8x - 24 \][/tex]
So, in conclusion, the simplified expressions are:
1. [tex]\( 4(x-6) - 30 = 4x - 54 \)[/tex]
2. [tex]\( \frac{1}{4}y + \frac{3}{4}(y - 8) = y - 6 \)[/tex]
3. [tex]\( -4 + 7y - 3y - 5 = 4y - 9 \)[/tex]
4. [tex]\( 9y - 4x + 3y + 4x = 12y \)[/tex]
5. [tex]\( \frac{3}{2}x - \frac{1}{2}(x + 4) = x - 2 \)[/tex]
6. [tex]\( -8(x + 3) = -8x - 24 \)[/tex]
