What is the difference between the expression (3x+4x)+2x

Question

What is the difference between the expression  (3x+4x)+2x
(
3
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+
4
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)
+
2
𝑥
  and the expression  3x+(4x+2x)
3
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+
(
4
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+
2
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)
 ?(1 point)
Responses

2x
2
𝑥
2 x

3x
3
𝑥
3 x

4x
4
𝑥
4 x

0

GPT-4o mini · Answer

To find the difference between the expressions $ (3x + 4x) + 2x $ and $ 3x + (4x + 2x) $, we can simplify both expressions.

For the first expression $ (3x + 4x) + 2x $:
- Simplify $ 3x + 4x $ to get $ 7x $.
- Then, $ 7x + 2x = 9x $.
For the second expression $ 3x + (4x + 2x) $:
- Simplify $ 4x + 2x $ to get $ 6x $.
- Then, $ 3x + 6x = 9x $.

Now, both expressions simplify to $ 9x $. Therefore, there is no difference between the two expressions.

The correct response is 0.

jimmy · Answer

Which of the following equations demonstrates the Associative Property?(1 point)
Responses

3x⋅(4x⋅5)=3x(4x+5)
3
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4
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=
3
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3 x times left parenthesis 4 x times 5 right parenthesis equals 3 x left parenthesis 4 x plus 5 right parenthesis

3x⋅(4x⋅5)=(3x⋅4x)⋅5
3
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4
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5
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=
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3
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⋅
5
3 x times left parenthesis 4 x times 5 right parenthesis equals left parenthesis 3 x times 4 x right parenthesis times 5

(3x+4x)⋅5=3x+(4x⋅5)
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4
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⋅
5
=
3
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+
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4
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⋅
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left parenthesis 3 x plus 4 x right parenthesis times 5 equals 3 x plus left parenthesis 4 x times 5 right parenthesis

3x⋅(4x+5x)=(3x+4x)⋅5

GPT-4o mini · Answer

The Associative Property states that the way in which numbers are grouped in addition or multiplication does not change their sum or product.

Given the options:

$3x \cdot (4x \cdot 5) = 3x(4x + 5)$ - This equation does not demonstrate the Associative Property; it mixes multiplication and addition incorrectly.
$3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5$ - This equation correctly shows the Associative Property of multiplication, where the grouping of factors can be changed without affecting the product.
$(3x + 4x) \cdot 5 = 3x + (4x \cdot 5)$ - This equation does not demonstrate the Associative Property; it does not correctly apply the property of addition or multiplication.
$3x \cdot (4x + 5x) = (3x + 4x) \cdot 5$ - This mixes multiplication and addition incorrectly as well, so it does not demonstrate the Associative Property.

The correct response that demonstrates the Associative Property is:

$3x \cdot (4x \cdot 5) = (3x \cdot 4x) \cdot 5$.

jimmy · Answer

Which expression is equivalent to (3r⋅5)⋅(−3⋅8)
(
3
𝑟
⋅
5
)
⋅
(
−
3
⋅
8
)
?(1 point)
Responses

−45r⋅5
−
45
𝑟
⋅
5
negative 45 r times 5

24r⋅(5⋅−3)
24
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⋅
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5
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−
3
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24 r times left parenthesis 5 times negative 3 right parenthesis

15r⋅(5⋅8)
15
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⋅
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5
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8
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15 r times left parenthesis 5 times 8 right parenthesis

(5⋅−3)⋅−9r

GPT-4o mini · Answer

To simplify the expression $(3r \cdot 5) \cdot (-3 \cdot 8)$ and find an equivalent expression, we first calculate each part separately: 1. **Calculate $3r \cdot 5$**: - This gives $15r$. 2. **Calculate $-3 \cdot 8$**: - This gives $-24$. Now we can combine these results: $$ (3r \cdot 5) \cdot (-3 \cdot 8) = 15r \cdot (-24) $$ This can be written as: $$ 15r \cdot -24 = -360r $$ Let's check each answer choice to see if any match $-360r$: 1. **−45r⋅5**: - This simplifies to $-225r$ (not equivalent). 2. **24r⋅(5⋅−3)**: - This simplifies to $24r \cdot (-15) = -360r$ (equivalent). 3. **15r⋅(5⋅8)**: - This simplifies to $15r \cdot 40 = 600r$ (not equivalent). 4. **(5⋅−3)⋅−9r**: - This simplifies to $-15 \cdot -9r = 135r$ (not equivalent). The expression that is equivalent to $(3r \cdot 5) \cdot (-3 \cdot 8)$ is: **24r⋅(5⋅−3)**.

jimmy · Answer

Use the Distributive Property to rewrite the expression (3x−5)(2x+2)
(
3
𝑥
−
5
)
(
2
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+
2
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.(1 point)
Responses

6x2−3x−10
6
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2
−
3
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−
10
6 x squared minus 3 x minus 10

x2−3x−10
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2
−
3
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−
10
x squared minus 3 x minus 10

6x2−4x−10
6
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2
−
4
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−
10
6 x squared minus 4 x minus 10

5x2−2x−3

GPT-4o mini · Answer

To rewrite the expression $(3x - 5)(2x + 2)$ using the Distributive Property, we will distribute each term in the first parenthesis by each term in the second parenthesis. 1. **Distributing $3x$**: - $3x \cdot 2x = 6x^2$ - $3x \cdot 2 = 6x$ 2. **Distributing $-5$**: - $-5 \cdot 2x = -10x$ - $-5 \cdot 2 = -10$ Now, combine all these results: $$ (3x - 5)(2x + 2) = 6x^2 + 6x - 10x - 10 $$ Combine like terms ($6x - 10x$): $$ 6x^2 - 4x - 10 $$ Thus, the expression rewritten is: **6x² - 4x - 10**. This corresponds to the third response option: **6x² − 4x − 10**.

jimmy · Answer

Use the Distributive Property to write the expression 4x2−36
4
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2
−
36
a different way.(1 point)
Responses

(x+18)(x−18)
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18
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−
18
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left parenthesis x plus 18 right parenthesis left parenthesis x minus 18 right parenthesis

4x(x−9)
4
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𝑥
−
9
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4 x left parenthesis x minus 9 right parenthesis

(2x+9)(x−9)
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2
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+
9
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(
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−
9
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left parenthesis 2 x plus 9 right parenthesis left parenthesis x minus 9 right parenthesis

(2x+6)(2x−6)

GPT-4o mini · Answer

To rewrite the expression $4x^2 - 36$ using the Distributive Property, we can first recognize that it can be factored as a difference of squares. The expression $4x^2 - 36$ can be rewritten as: $$ 4(x^2 - 9) $$ Next, $x^2 - 9$ is also a difference of squares and can be factored as: $$ x^2 - 9 = (x + 3)(x - 3) $$ Putting it all together, we can express $4x^2 - 36$ as: $$ 4(x^2 - 9) = 4((x + 3)(x - 3)) = (2(x + 3))(2(x - 3)) = (2x + 6)(2x - 6) $$ Now, let's compare this with the given options: 1. **$(x + 18)(x - 18)$** - This is incorrect; it doesn't match our expression. 2. **$4x(x - 9)$** - This is also incorrect; it doesn't match our expression. 3. **$(2x + 9)(x - 9)$** - This does not match our expression. 4. **$(2x + 6)(2x - 6)$** - This matches our expression. Thus, the correct answer is: **$(2x + 6)(2x - 6)$**.

jimmy · Answer

Find the expression that is equivalent to 3x2+2x−5x−2x2+2
3
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2
+
2
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−
5
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−
2
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2
+
2
.(1 point)
Responses

(x−2)(x−1)
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2
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(
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1
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left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis

(3x+2)(x+1)
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2
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left parenthesis 3 x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis

(x+1)(x+2)
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1
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left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis

(x+2)(x−2)

GPT-4o mini · Answer

To find an expression equivalent to $3x^2 + 2x - 5x - 2x^2 + 2$, we can first combine like terms: 1. Combine the $x^2$ terms: $$ 3x^2 - 2x^2 = x^2 $$ 2. Combine the $x$ terms: $$ 2x - 5x = -3x $$ 3. The constant term is: $$ +2 $$ Now we can rewrite the entire expression with the simplified terms: $$ x^2 - 3x + 2 $$ Next, we will factor the quadratic expression $x^2 - 3x + 2$: We need two numbers that multiply to $2$ (the constant term) and add to $-3$ (the coefficient of $x$). The numbers $-1$ and $-2$ satisfy these conditions: $$ (x - 1)(x - 2) $$ Now let's compare this factored form with the given options: 1. **$(x - 2)(x - 1)$** - This is equivalent to $x^2 - 3x + 2$. 2. **$(3x + 2)(x + 1)$** - This does not match our expression. 3. **$(x + 1)(x + 2)$** - This does not match our expression. 4. **$(x + 2)(x - 2)$** - This does not match our expression. Thus, the correct answer is: **$(x - 2)(x - 1)$**.