Select The Correct Answer.What Is The Solution To $|x-7| \geq 1$?A. $6 \leq X \leq 8$B. $-8 \leq X \leq -6$C. $x \geq -6$ Or $x \leq -8$D. $x \geq 8$ Or $x \leq 6$

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Introduction

Absolute value inequalities are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value functions. In this article, we will focus on solving the inequality ∣x−7∣≥1|x-7| \geq 1, which is a classic example of an absolute value inequality. We will break down the solution step by step, and by the end of this article, you will be able to solve similar inequalities with ease.

Understanding Absolute Value Functions

Before we dive into solving the inequality, let's quickly review the properties of absolute value functions. The absolute value function, denoted by ∣x∣|x|, is defined as:

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

In other words, the absolute value of a number is its distance from zero on the number line. For example, ∣5∣=5|5| = 5 and ∣−3∣=3|-3| = 3.

Solving the Inequality

Now that we have a good understanding of absolute value functions, let's solve the inequality ∣x−7∣≥1|x-7| \geq 1. To do this, we need to consider two cases:

Case 1: x−7≥0x-7 \geq 0

In this case, the absolute value function simplifies to ∣x−7∣=x−7|x-7| = x-7. We can now rewrite the inequality as:

x−7≥1x-7 \geq 1

Solving for xx, we get:

x≥8x \geq 8

Case 2: x−7<0x-7 < 0

In this case, the absolute value function simplifies to ∣x−7∣=−(x−7)|x-7| = -(x-7). We can now rewrite the inequality as:

−(x−7)≥1-(x-7) \geq 1

Simplifying, we get:

−x+7≥1-x+7 \geq 1

Solving for xx, we get:

x≤6x \leq 6

Combining the Solutions

We have now solved the inequality for both cases. In Case 1, we found that x≥8x \geq 8, and in Case 2, we found that x≤6x \leq 6. However, we need to consider the fact that the absolute value function is defined as ∣x−7∣≥1|x-7| \geq 1. This means that the solution must be true for both cases.

Therefore, the correct solution is:

x≥8 or x≤6x \geq 8 \text{ or } x \leq 6

Conclusion

Solving absolute value inequalities requires a clear understanding of the properties of absolute value functions. By breaking down the solution into two cases and considering the definition of the absolute value function, we can arrive at the correct solution. In this article, we solved the inequality ∣x−7∣≥1|x-7| \geq 1 and found that the correct solution is x≥8x \geq 8 or x≤6x \leq 6. We hope that this article has provided you with a clear understanding of how to solve absolute value inequalities.

Answer

The correct answer is:

D. x≥8x \geq 8 or x≤6x \leq 6

Discussion

Do you have any questions or comments about solving absolute value inequalities? Share your thoughts in the discussion section below!

Discussion Section

  • Question 1: How do you solve absolute value inequalities?
  • Answer 1: To solve absolute value inequalities, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You then solve the inequality for each case and combine the solutions.
  • Question 2: What is the definition of the absolute value function?
  • Answer 2: The absolute value function is defined as ∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}.
  • Question 3: How do you know which case to use when solving an absolute value inequality?
  • Answer 3: You need to consider the sign of the expression inside the absolute value. If it is positive, you use Case 1. If it is negative, you use Case 2.

Related Topics

  • Solving Linear Inequalities: Linear inequalities are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of linear functions.
  • Solving Quadratic Inequalities: Quadratic inequalities are a type of inequality that involves a quadratic expression. Solving them requires a clear understanding of the properties of quadratic functions.
  • Solving Systems of Inequalities: Systems of inequalities are a set of inequalities that involve multiple variables. Solving them requires a clear understanding of the properties of linear and quadratic functions.

References

  • Khan Academy: Khan Academy is a free online platform that provides video lessons and practice exercises on a wide range of topics, including mathematics.
  • Mathway: Mathway is a free online platform that provides step-by-step solutions to a wide range of mathematical problems, including absolute value inequalities.
  • Wolfram Alpha: Wolfram Alpha is a free online platform that provides step-by-step solutions to a wide range of mathematical problems, including absolute value inequalities.
    Absolute Value Inequality Q&A =============================

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves an absolute value expression. It is a mathematical statement that compares the absolute value of an expression to a constant or another expression.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You then solve the inequality for each case and combine the solutions.

Q: What is the definition of the absolute value function?

A: The absolute value function is defined as ∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}.

Q: How do I know which case to use when solving an absolute value inequality?

A: You need to consider the sign of the expression inside the absolute value. If it is positive, you use Case 1. If it is negative, you use Case 2.

Q: What is the difference between an absolute value inequality and a linear inequality?

A: An absolute value inequality involves an absolute value expression, while a linear inequality involves a linear expression. Absolute value inequalities are more complex and require a different approach to solve.

Q: Can I use algebraic methods to solve absolute value inequalities?

A: Yes, you can use algebraic methods to solve absolute value inequalities. However, you need to be careful when simplifying the inequality and combining the solutions.

Q: How do I graph an absolute value inequality?

A: To graph an absolute value inequality, you need to graph the absolute value function and then shade the region that satisfies the inequality.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

  • Not considering both cases when solving the inequality
  • Not simplifying the inequality correctly
  • Not combining the solutions correctly
  • Not checking the solutions in the original inequality

Q: How do I check my solutions to an absolute value inequality?

A: To check your solutions to an absolute value inequality, you need to plug the solutions back into the original inequality and verify that they satisfy the inequality.

Q: What are some real-world applications of absolute value inequalities?

A: Absolute value inequalities have many real-world applications, including:

  • Physics: to model the motion of objects
  • Engineering: to design and optimize systems
  • Economics: to model economic systems and make predictions
  • Computer Science: to solve problems in computer graphics and game development

Q: Can I use technology to solve absolute value inequalities?

A: Yes, you can use technology to solve absolute value inequalities. Many graphing calculators and computer algebra systems can solve absolute value inequalities and provide step-by-step solutions.

Q: How do I choose the best method to solve an absolute value inequality?

A: To choose the best method to solve an absolute value inequality, you need to consider the complexity of the inequality and the tools available to you. If the inequality is simple, you can use algebraic methods. If the inequality is complex, you may need to use technology or graphing methods.

Q: What are some common types of absolute value inequalities?

A: Some common types of absolute value inequalities include:

  • ∣x∣≥a|x| \geq a
  • ∣x∣≤a|x| \leq a
  • ∣x−a∣≥b|x-a| \geq b
  • ∣x−a∣≤b|x-a| \leq b

Q: How do I solve a system of absolute value inequalities?

A: To solve a system of absolute value inequalities, you need to solve each inequality separately and then combine the solutions. You can use algebraic methods or graphing methods to solve the system.

Q: What are some tips for solving absolute value inequalities?

A: Some tips for solving absolute value inequalities include:

  • Read the problem carefully and understand what is being asked
  • Use algebraic methods or graphing methods to solve the inequality
  • Check your solutions in the original inequality
  • Use technology to check your solutions and provide step-by-step solutions

Q: How do I know if I have solved an absolute value inequality correctly?

A: To know if you have solved an absolute value inequality correctly, you need to check your solutions in the original inequality and verify that they satisfy the inequality. You can also use technology to check your solutions and provide step-by-step solutions.