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# Bài 36 trang 51 SGK Toán 8 Tập 2 – VietJack.com

## Introduction

In this article, we will explore the process of solving absolute value equations. We will go through several examples and provide detailed explanations step by step. By the end of this article, you will have a solid understanding of how to solve these types of equations.

## Understanding Absolute Value Equations

Absolute value equations contain the absolute value symbol, represented by two vertical lines ( | | ). These equations can have multiple solutions, which makes them interesting to solve. Let’s dive into some examples to see how it works.

### Example 1: |2x| = x – 6

To begin, let’s analyze the possible values of x based on the positive and negative cases of the absolute value.

• When 2x ≥ 0 (i.e., x ≥ 0), |2x| simplifies to 2x.
• When 2x < 0 (i.e., x < 0), |2x| simplifies to -2x.

Now, let’s analyze the equation based on these cases:

1. Case 1: 2x = x – 6 (x ≥ 0)

• Simplifying the equation, we get x = -6.
• However, this solution does not satisfy the condition x ≥ 0, so it is not valid.
2. Case 2: -2x = x – 6 (x < 0)

• Simplifying the equation, we get x = 2.
• This solution satisfies the condition x < 0, so it is a valid solution.

Therefore, the equation |2x| = x – 6 has only one solution, x = 2.

### Example 2: |-3x| = x – 8

Let’s apply the same analysis to this equation:

• When -3x ≥ 0 (i.e., x ≤ 0), |-3x| simplifies to -3x.
• When -3x < 0 (i.e., x > 0), |-3x| simplifies to 3x.

Now, let’s analyze the equation based on these cases:

1. Case 1: -3x = x – 8 (x ≤ 0)

• Simplifying the equation, we get x = 2.
• However, this solution does not satisfy the condition x ≤ 0, so it is not valid.
2. Case 2: 3x = x – 8 (x > 0)

• Simplifying the equation, we get x = -4.
• This solution satisfies the condition x > 0, so it is a valid solution.

Therefore, the equation |-3x| = x – 8 has only one solution, x = -4.

### Example 3: |4x| = 2x + 12

Let’s apply the same analysis once again:

• When 4x ≥ 0 (i.e., x ≥ 0), |4x| simplifies to 4x.
• When 4x < 0 (i.e., x < 0), |4x| simplifies to -4x.

Now, let’s analyze the equation based on these cases:

1. Case 1: 4x = 2x + 12 (x ≥ 0)

• Simplifying the equation, we get x = 6.
• This solution satisfies the condition x ≥ 0, so it is a valid solution.
2. Case 2: -4x = 2x + 12 (x < 0)

• Simplifying the equation, we get x = -2.
• This solution satisfies the condition x < 0, so it is a valid solution.

Therefore, the equation |4x| = 2x + 12 has two solutions, x = 6 and x = -2.

### Example 4: |-5x| – 16 = 3x

Finally, let’s analyze this equation:

• When -5x ≥ 0 (i.e., x ≤ 0), |-5x| simplifies to -5x.
• When -5x < 0 (i.e., x > 0), |-5x| simplifies to 5x.

Now, let’s analyze the equation based on these cases:

1. Case 1: -5x – 16 = 3x (x ≤ 0)

• Simplifying the equation, we get x = -2.
• This solution satisfies the condition x ≤ 0, so it is a valid solution.
2. Case 2: 5x – 16 = 3x (x > 0)

• Simplifying the equation, we get x = 8.
• This solution satisfies the condition x > 0, so it is a valid solution.

Therefore, the equation |-5x| – 16 = 3x has two solutions, x = -2 and x = 8.

## Conclusion

In this article, we explored the process of solving absolute value equations. We examined several examples and discussed the steps involved in finding the solutions. By following the analysis we provided, you will be able to confidently solve these types of equations on your own. Remember to pay attention to the conditions of x to ensure you find valid solutions. For more math-related articles, visit Kienthucykhoa.com.

## References

### Kiến Thức Y Khoa

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