Math Problem Statement

-5∣5x−1∣−6= −41

Solution

We are given the equation:

55x16=41-5|5x - 1| - 6 = -41

Step 1: Isolate the absolute value term

We first add 6 to both sides:

55x1=41+6-5|5x - 1| = -41 + 6 55x1=35-5|5x - 1| = -35

Step 2: Divide both sides by -5

To isolate the absolute value expression, divide both sides by -5:

5x1=355|5x - 1| = \frac{-35}{-5} 5x1=7|5x - 1| = 7

Step 3: Solve the absolute value equation

The expression 5x1=7|5x - 1| = 7 has two cases:

  1. 5x1=75x - 1 = 7
  2. 5x1=75x - 1 = -7

Case 1: 5x1=75x - 1 = 7

Add 1 to both sides: 5x=85x = 8 Divide by 5: x=85=1.6x = \frac{8}{5} = 1.6

Case 2: 5x1=75x - 1 = -7

Add 1 to both sides: 5x=65x = -6 Divide by 5: x=65=1.2x = \frac{-6}{5} = -1.2

Final Solution:

The solutions are: x=1.6orx=1.2x = 1.6 \quad \text{or} \quad x = -1.2

Would you like any further details or explanations?

Here are five related questions for practice:

  1. Solve 3x+2=10|3x + 2| = 10.
  2. Solve 24x3+1=92|4x - 3| + 1 = 9.
  3. What is the general process for solving absolute value equations?
  4. How does isolating the absolute value term help in solving such equations?
  5. What happens when an absolute value equation has no solution?

Tip: When solving absolute value equations, always remember to split the equation into two cases—one for the positive and one for the negative scenario!

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Equations
Linear Equations

Formulas

Absolute value equation: |Ax + B| = C
Linear equation: Ax + B = C

Theorems

Absolute value equation solving method

Suitable Grade Level

Grades 8-10