What is the solution for the following absolute value inequality?

When faced with an absolute value inequality, it is important to understand its properties and how to find the solution. Absolute value inequalities involve the absolute value of a variable, and the solution consists of all the possible values that satisfy the given inequality. Let’s explore the steps to solve an absolute value inequality and find the solution to a specific example.

To illustrate the process, let’s consider the absolute value inequality: |3x – 2| < 10. Our goal is to determine the values of x that satisfy this inequality.

Solving an Absolute Value Inequality

To solve an absolute value inequality, follow these steps:

1. Isolate the absolute value expression: In our example, we have |3x – 2| < 10. Isolating the absolute value gives us the inequality 3x - 2 < 10. 2. Set up two cases: Since the absolute value of a number can be either positive or negative, we need to consider two scenarios for the inequality. In our case, we set up two cases: 3x – 2 < 10 and -(3x - 2) < 10. 3. Solve each case: Solving the first case, 3x – 2 < 10, we add 2 to both sides and get 3x < 12. Dividing both sides by 3, we find x < 4. 4. Solving the second case: For -(3x – 2) < 10, we start by distributing the negative sign inside the absolute value: -3x + 2 < 10. Subtracting 2 from both sides, we obtain -3x < 8. Dividing both sides by -3, we get x > -8/3.

The Solution

Now that we have found the solutions for both cases, we need to determine the final solution by considering the intersection of these two cases.

The solution for the absolute value inequality |3x – 2| < 10 is: x < 4 and x > -8/3.

Therefore, the solution set for this inequality is (-8/3, 4).

Frequently Asked Questions (FAQs)

1. What is an absolute value?

The absolute value of a number is its distance from zero on the number line. It is always a non-negative value.

2. When do we use absolute value inequalities?

Absolute value inequalities are used when we need to find a range of values that satisfy a given condition involving absolute value.

3. How is the absolute value represented in equations and inequalities?

The absolute value of a number x is typically represented as |x|.

4. What does it mean for an absolute value inequality to be less than or greater than a constant?

If an absolute value inequality is less than a constant, it means that the absolute value expression is closer to zero than the constant. If it is greater than a constant, it means the absolute value expression is farther from zero than the constant.

5. Can an absolute value inequality have multiple solutions?

Yes, absolute value inequalities can have multiple solutions, as they often represent a range of values that satisfy the given condition.

6. How does the direction of the inequality affect the solution set?

The direction of the inequality determines whether the solution set includes values greater than or less than the given constant.

7. Can we have an absolute value inequality with a negative constant?

Yes, absolute value inequalities can involve negative constants. The negative sign only affects the direction of the inequality and does not affect the procedure used to solve the inequality.

8. Is it necessary to consider two cases when solving an absolute value inequality?

Yes, since the absolute value of a number can be positive or negative, we need to consider both cases in order to find the complete solution.

9. How can we represent the solution set for an absolute value inequality?

The solution set for an absolute value inequality is typically shown using interval notation, where parentheses or brackets indicate whether the endpoints are included or excluded.

10. Are there any shortcuts or tricks to solve absolute value inequalities?

While there might be specific strategies for certain absolute value inequalities, it is generally recommended to follow the general steps outlined earlier to find the solution.

11. Can absolute value inequalities be applied in real-life situations?

Absolutely! Absolute value inequalities are applied in various fields such as physics, economics, and engineering to model real-life situations involving constraints or limitations.

12. Do absolute value inequalities appear in other areas of mathematics?

Yes, absolute value inequalities are encountered in algebra, calculus, and other branches of mathematics because they are a fundamental concept used to solve equations and inequalities accurately.

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