Question

1 noah says that $9x - 2x + 4x$ is equivalent to $3x$, because the subtraction sign tells us to subtract everything that comes after $9x$. bena says that $9x - 2x + 4x$ is equivalent to $11x$, because the subtraction only applies to $2x$. do you agree with either of them? explain your reasoning. 2 identify the error in generating an expression equivalent to $4 + 2x - \frac{1}{2}(10 - 4x)$. then correct the error. $4 + 2x + \frac{-1}{2}(10 + -4x)$ $4 + 2x + -5 + 2x$ $4 + 2x - 5 + 2x$ $-1$ 3 select all expressions that are equivalent to $5x - 15 - 20x + 10$. a $5x - (15 + 20x) + 10$ b $5x + -15 + -20x + 10$ c $5(x - 3 - 4x + 2)$ d $-5(-x + 3 + 4x + -2)$ e $-15x - 5$ f $-5(3x + 0)$ g $-15\left(x - \frac{1}{3}\
ight)$

Explanation:

Problem 1

Step1: Combine like terms left to right

$9x - 2x + 4x = 7x + 4x$

Step2: Calculate final sum of terms

$7x + 4x = 11x$

Step1: Identify the sign error

The original expression is $4 + 2x - \frac{1}{2}(10 - 4x)$. The error incorrectly rewrote $-\frac{1}{2}(10 - 4x)$ as $+\frac{1}{2}(10 + -4x)$ (unnecessary sign rewrite) and then failed to combine like terms properly at the end.

Step2: Distribute the $-\frac{1}{2}$ correctly

$4 + 2x - \frac{1}{2}(10) + \frac{1}{2}(4x) = 4 + 2x - 5 + 2x$

Step3: Combine constant and variable terms

$(4 - 5) + (2x + 2x) = -1 + 4x$

Step1: Simplify the target expression

$5x - 15 - 20x + 10 = (5x - 20x) + (-15 + 10) = -15x - 5$

Step2: Simplify each option

Option A:

$5x - (15 + 20x) + 10 = 5x -15 -20x +10 = -15x -5$

Option B:

$5x + -15 + -20x +10 = (5x-20x)+(-15+10) = -15x -5$

Option C:

$5(x - 3 - 4x + 2) = 5(-3x -1) = -15x -5$

Option D:

$-5(-x + 3 + 4x + -2) = -5(3x +1) = -15x -5$

Option E:

$-15x -5$ (matches directly)

Option F:

$-5(3x +1) = -15x -5$

Option G:

$-15(x - \frac{1}{3}) = -15x +5$ (does not match)

Answer:

I agree with Elena. Noah incorrectly applied the subtraction sign to all terms after $9x$, while Elena correctly only subtracted $2x$ before adding $4x$, resulting in $11x$.

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Problem 2