Math Problem Statement

solve

|3x - 4| = 8 - x

Solution

To solve the equation:

3x4=8x|3x - 4| = 8 - x

We need to consider two cases because of the absolute value:

Case 1: 3x4=8x3x - 4 = 8 - x

In this case, we can remove the absolute value and solve for xx:

3x4=8x3x - 4 = 8 - x

  1. Add xx to both sides to get rid of the x-x on the right-hand side: 4x4=84x - 4 = 8

  2. Add 4 to both sides to isolate the term with xx: 4x=124x = 12

  3. Divide both sides by 4 to solve for xx: x=3x = 3

Case 2: 3x4=(8x)3x - 4 = -(8 - x)

In this case, remove the absolute value by setting the expression equal to the negative of the right-hand side:

3x4=(8x)3x - 4 = -(8 - x)

Simplify the right-hand side: 3x4=8+x3x - 4 = -8 + x

  1. Subtract xx from both sides: 2x4=82x - 4 = -8

  2. Add 4 to both sides: 2x=42x = -4

  3. Divide both sides by 2 to solve for xx: x=2x = -2

Verifying solutions:

For x=3x = 3: 3(3)4=8394=55=5(True)|3(3) - 4| = 8 - 3 \quad \Rightarrow \quad |9 - 4| = 5 \quad \Rightarrow \quad 5 = 5 \quad (\text{True})

For x=2x = -2: 3(2)4=8(2)64=1010=10(True)|3(-2) - 4| = 8 - (-2) \quad \Rightarrow \quad |-6 - 4| = 10 \quad \Rightarrow \quad | -10 | = 10 \quad (\text{True})

Final Solutions:

x=3andx=2x = 3 \quad \text{and} \quad x = -2

Let me know if you would like more details or if you have any questions.

Here are 5 related questions for practice:

  1. Solve 2x+1=7x|2x + 1| = 7 - x.
  2. Solve x+5=3x2|x + 5| = 3x - 2.
  3. Solve 4x7=9|4x - 7| = 9.
  4. Solve 5x+2=2x+8|5x + 2| = 2x + 8.
  5. Solve x3=42x|x - 3| = 4 - 2x.

Tip: When solving absolute value equations, always split the equation into two cases: one for the positive expression and one for the negative expression.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Equations

Formulas

|A| = B → A = B or A = -B

Theorems

Absolute Value Property

Suitable Grade Level

Grades 8-10