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What are the values of $𝑥$ that satisfy the equation | $x-7$ | $-3$ | $x-2$ | $+4$ | $x+8$ | $+$ | $x$ | $=$ $441$ ?

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Identify biased samples

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Q. What are the values of

𝑥

that satisfy the equation |

x-7

|

-3

|

x-2

|

+4

|

x+8

|

+

|

x

|

=

441

?

Understand Absolute Value Properties: Understand the properties of absolute value. Absolute value equations can have two possible cases for each absolute value expression: one where the inside is non-negative (and the absolute value acts like the identity function), and one where the inside is negative (and the absolute value acts like the negation function). We will need to consider these cases to solve the equation.
Set Up Critical Point Cases: Set up different cases based on the critical points of the absolute value expressions. $\newline$ The critical points are $x = 7$ , $x = 2$ , $x = -8$ , and $x = 0$ . These points divide the number line into intervals where the expressions inside the absolute values do not change sign. We will need to solve the equation in each interval separately.
Solve Equations in Intervals: Solve the equation in each interval. $\newline$ We will start with the interval $x > 7$ , where all the expressions inside the absolute values are non-negative. The equation simplifies to: $\newline$ $x - 7 - 3(x - 2) + 4(x + 8) + x = 441$
Simplify Equation for $x > 7$ : Simplify the equation for the interval $x > 7$ . Combine like terms: $x - 7 - 3x + 6 + 4x + 32 + x = 441$ $(1 - 3 + 4 + 1)x + (6 - 7 + 32) = 441$ $3x + 31 = 441$
Solve for $x$ in $x > 7$ : Solve for $x$ in the interval $x > 7$ .
Subtract $31$ from both sides:
$3x = 441 - 31$
$3x = 410$
Divide by $3$ :
$x = \frac{410}{3}$
$x \approx 136.67$
Check Solution in $x > 7$ : Check if the solution $x \approx 136.67$ is in the interval $x > 7$ . $\newline$ Since $136.67$ is greater than $7$ , it is in the interval we are considering.
Repeat Steps for Other Intervals: Repeat steps $3$ to $6$ for the other intervals ( $x$ between $2$ and $7$ , $x$ between $-8$ and $2$ , $x$ between $0$ and $-8$ , and $x < -8$ ). $\newline$ However, since we have already found a solution in the interval $2$ $0$ , we need to check if there are any other solutions in the other intervals. If there are, we will include them in our final answer.
Verify Solution in Original Equation: Verify if the solution $x \approx 136.67$ satisfies the original equation. $\newline$ Plug $x \approx 136.67$ back into the original equation: $\newline$ $|136.67 - 7| - 3|136.67 - 2| + 4|136.67 + 8| + |136.67| = 441$ $\newline$ $129.67 - 3(134.67) + 4(144.67) + 136.67 = 441$ $\newline$ $129.67 - 404.01 + 578.68 + 136.67 \approx 441$ $\newline$ The left side does not equal $441$ , which means there is a math error in our previous steps.

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