Math Problem Statement

How do I finish solving this

How do I finish solving this?

Solution

To finish solving the equation, let's break it down step by step:

The equation is: 2+106a+3=322 + 10 | -6a + 3 | = 32 You've simplified it by dividing both sides by 10: 6a+3=3| -6a + 3 | = 3

Now, to solve absolute value equations, you must consider both the positive and negative possibilities of the expression inside the absolute value. So, we need to set up two equations:

  1. 6a+3=3-6a + 3 = 3 (when the expression inside the absolute value is positive)
  2. 6a+3=3-6a + 3 = -3 (when the expression inside the absolute value is negative)

Let's solve these two equations one by one.

Equation 1: 6a+3=3-6a + 3 = 3

Subtract 3 from both sides: 6a=0-6a = 0 Now, divide by -6: a=0a = 0

Equation 2: 6a+3=3-6a + 3 = -3

Subtract 3 from both sides: 6a=6-6a = -6 Now, divide by -6: a=1a = 1

Final Answer:

The solutions are a=0a = 0 and a=1a = 1.

Tip:

Always remember to check your solutions by substituting them back into the original equation to verify that they satisfy the equation. This step ensures you haven’t made any mistakes in the process.

Would you like to go over how to check these solutions, or do you have any questions about the process? Here's a list of related questions to explore further:

  1. What happens if the absolute value expression has more complex terms inside?
  2. How can we solve equations involving absolute value and inequalities?
  3. What are some common mistakes when solving absolute value equations?
  4. How can we graph absolute value functions?
  5. What is the difference between solving absolute value equations and inequalities?

Let me know if you'd like to explore any of these!

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Equations
Linear Equations

Formulas

|A| = B has two cases: A = B or A = -B

Theorems

Absolute Value Property

Suitable Grade Level

Grades 8-10