expanding expressions
expand each expression. combine like terms where possible.
1 ( 4(x - 2) )
2 ( -3(x + 7) )
3 ( -4(-x - 8) )
4 ( \frac{1}{3}(x - 9) )
5 ( -\frac{1}{4}(x + 16) )
6 ( -\frac{1}{5}(-x - 35) )
7 ( \frac{2}{3}(x + 18 - 2x) )
8 ( \frac{3}{4}(16x - 27 - 1) )
9 ( -12left( \frac{5}{6}x - 5
ight) + 2x )
determine which expressions, if any, are equivalent. show your work.
10 ( 4(x - 3) )
( 6x - 2(x - 3) )
( x + 3(x - 2) - 6 )

Question

Accepted Answer

### Problem 1: $ 4(x - 2) $
# Explanation:
## Step 1: Apply the distributive property $ a(b - c)=ab - ac $
$ 4(x - 2)=4\times x-4\times2 $
## Step 2: Simplify the multiplication
$ 4x - 8 $
# Answer: $ 4x - 8 $

### Problem 2: $ -3(x + 7) $
# Explanation:
## Step 1: Apply the distributive property $ a(b + c)=ab + ac $
$ -3(x + 7)=-3\times x+(-3)\times7 $
## Step 2: Simplify the multiplication
$ -3x - 21 $
# Answer: $ -3x - 21 $

### Problem 3: $ -4(-x - 8) $
# Explanation:
## Step 1: Apply the distributive property $ a(-b - c)=-ab - ac $, here $ a = - 4,b=x,c = 8 $
$ -4(-x - 8)=(-4)\times(-x)+(-4)\times(-8) $
## Step 2: Simplify the multiplication
$ 4x + 32 $ (Note: The original handwritten answer seems incorrect, the correct expansion is $ 4x + 32 $)
# Answer: $ 4x + 32 $

### Problem 4: $ \frac{1}{3}(x - 9) $
# Explanation:
## Step 1: Apply the distributive property $ a(b - c)=ab - ac $
$ \frac{1}{3}(x - 9)=\frac{1}{3}\times x-\frac{1}{3}\times9 $
## Step 2: Simplify the multiplication
$ \frac{1}{3}x-3 $
# Answer: $ \frac{1}{3}x - 3 $

### Problem 5: $ -\frac{1}{4}(x + 16) $
# Explanation:
## Step 1: Apply the distributive property $ a(b + c)=ab + ac $
$ -\frac{1}{4}(x + 16)=-\frac{1}{4}\times x+(-\frac{1}{4})\times16 $
## Step 2: Simplify the multiplication
$ -\frac{1}{4}x - 4 $
# Answer: $ -\frac{1}{4}x - 4 $

### Problem 6: $ -\frac{1}{5}(-x - 35) $
# Explanation:
## Step 1: Apply the distributive property $ a(-b - c)=-ab - ac $, here $ a=-\frac{1}{5},b = x,c = 35 $
$ -\frac{1}{5}(-x - 35)=(-\frac{1}{5})\times(-x)+(-\frac{1}{5})\times(-35) $
## Step 2: Simplify the multiplication
$ \frac{1}{5}x + 7 $
# Answer: $ \frac{1}{5}x + 7 $

### Problem 7: $ \frac{2}{3}(x + 18 - 2x) $
# Explanation:
## Step 1: Combine like terms inside the parentheses
$ x-2x+18=-x + 18 $
## Step 2: Apply the distributive property $ a(b + c)=ab+ac $
$ \frac{2}{3}(-x + 18)=\frac{2}{3}\times(-x)+\frac{2}{3}\times18 $
## Step 3: Simplify the multiplication
$ -\frac{2}{3}x + 12 $
# Answer: $ -\frac{2}{3}x + 12 $

### Problem 8: $ \frac{3}{4}(16x - 27 - 1) $
# Explanation:
## Step 1: Combine like terms inside the parentheses
$ 16x-(27 + 1)=16x-28 $
## Step 2: Apply the distributive property $ a(b - c)=ab - ac $
$ \frac{3}{4}(16x - 28)=\frac{3}{4}\times16x-\frac{3}{4}\times28 $
## Step 3: Simplify the multiplication
$ 12x-21 $
# Answer: $ 12x - 21 $

### Problem 9: $ -12(\frac{5}{6}x - 5)+2x $
# Explanation:
## Step 1: Apply the distributive property $ a(b - c)=ab - ac $
$ -12\times\frac{5}{6}x-12\times(-5)+2x $
## Step 2: Simplify the multiplication
$ - 10x + 60+2x $
## Step 3: Combine like terms
$ (-10x + 2x)+60=-8x + 60 $
# Answer: $ -8x + 60 $

### Problem 10: Determine which expressions are equivalent: $ 4(x - 3) $, $ 6x - 2(x - 3) $, $ x + 3(x - 2)-6 $
# Explanation:
#### For $ 4(x - 3) $:
## Step 1: Apply the distributive property
$ 4(x - 3)=4x-12 $

#### For $ 6x - 2(x - 3) $:
## Step 1: Apply the distributive property $ a(b - c)=ab - ac $
$ 6x-2x + 6 $
## Step 2: Combine like terms
$ 4x + 6 $ (Wait, correction: $ 6x-2x + 6=4x + 6 $? No, wait $ -2(x - 3)=-2x + 6 $, so $ 6x-2x + 6 = 4x+6 $? Wait, no, let's re - do:

Wait, $ 6x-2(x - 3)=6x-(2x - 6)=6x - 2x+6 = 4x + 6 $? No, that's wrong. Wait, $ -2(x - 3)=-2x+6 $, so $ 6x-2x + 6=4x + 6 $? Wait, no, the correct expansion:

$ 6x-2(x - 3)=6x-2x + 6=4x + 6 $ (Wait, but let's check the third expression)

#### For $ x + 3(x - 2)-6 $:
## Step 1: Apply the distributive property $ a(b - c)=ab - ac $
$ x+3x-6 - 6 $
## Step 2: Combine like terms
$ (x + 3x)+(-6-6)=4x-12 $

Wait, there was a mistake in the second expression. Let's re - expand $ 6x - 2(x - 3) $:

$ 6x-2(x - 3)=6x-2x + 6=4x + 6 $? No, no, $ -2(x - 3)=-2x + 6 $, so $ 6x-2x+6 = 4x + 6 $. But the first expression is $ 4x-12 $ and the third expression:

$ x + 3(x - 2)-6=x + 3x-6 - 6=4x-12 $

Wait, so $ 4(x - 3)=4x - 12 $ and $ x + 3(x - 2)-6=4x-12 $, and $ 6x - 2(x - 3)=4x + 6 $

So the equivalent expressions are $ 4(x - 3) $ and $ x + 3(x - 2)-6 $

# Answer:
- $ 4(x - 3)=4x - 12 $
- $ 6x - 2(x - 3)=4x + 6 $
- $ x + 3(x - 2)-6=4x - 12 $

The expressions $ 4(x - 3) $ and $ x + 3(x - 2)-6 $ are equivalent.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( 4(x - 2) \)

Step 1: Apply the distributive property \( a(b - c)=ab - ac \)

Step 2: Simplify the multiplication

Step 1: Apply the distributive property \( a(b + c)=ab + ac \)

Step 2: Simplify the multiplication

Step 1: Apply the distributive property \( a(-b - c)=-ab - ac \), here \( a = - 4,b=x,c = 8 \)

Step 2: Simplify the multiplication

Answer:

Problem 2: \( -3(x + 7) \)