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

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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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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. 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. We will start with the interval x > 7, where all the expressions inside the absolute values are non-negative. The equation simplifies to: x - 7 - 3(x - 2) + 4(x + 8) + x = 441Simplify 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 = 441Solve 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 = 410 / 3 x ≈ 136.67Check Solution in x > 7: Check if the solution x ≈ 136.67 is in the interval x > 7. 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). However, since we have already found a solution in the interval x > 7, 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 ≈ 136.67 satisfies the original equation. Plug x ≈ 136.67 back into the original equation: |136.67 - 7| - 3|136.67 - 2| + 4|136.67 + 8| + |136.67| = 441 129.67 - 3(134.67) + 4(144.67) + 136.67 = 441 129.67 - 404.01 + 578.68 + 136.67 ≈ 441 The left side does not equal 441, which means there is a math error in our previous steps.

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

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What are the values of 𝑥𝑥x that satisfy the equation |x−7x-7x−7|−3-3−3|x−2x-2x−2|+4+4+4|x+8x+8x+8|+++|xxx|===441441441 ?

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