Solve The Absolute Value Inequality: $|x+12|+5\ \textless \ 27$.Isolate The Absolute Value By Subtracting 5 From Both Sides.

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Introduction

Absolute value inequalities are a type of mathematical problem that involves solving equations with absolute values. In this article, we will focus on solving the absolute value inequality $|x+12|+5 < 27$. We will start by isolating the absolute value by subtracting 5 from both sides, and then proceed to solve the inequality.

Isolating the Absolute Value

To isolate the absolute value, we need to subtract 5 from both sides of the inequality. This will give us $|x+12| < 22$. The absolute value is now isolated, and we can proceed to solve the inequality.

Solving the Inequality

To solve the inequality, we need to consider two cases: when $x+12$ is positive, and when $x+12$ is negative.

Case 1: $x+12 \geq 0$

When $x+12 \geq 0$, the absolute value $|x+12|$ can be rewritten as $x+12$. The inequality becomes $x+12 < 22$. To solve for $x$, we need to subtract 12 from both sides of the inequality. This gives us $x < 10$. Therefore, when $x+12 \geq 0$, the solution to the inequality is $x < 10$.

Case 2: $x+12 < 0$

When $x+12 < 0$, the absolute value $|x+12|$ can be rewritten as $-x-12$. The inequality becomes $-x-12 < 22$. To solve for $x$, we need to add 12 to both sides of the inequality, and then multiply both sides by -1. This gives us $x > -34$. Therefore, when $x+12 < 0$, the solution to the inequality is $x > -34$.

Combining the Solutions

We have found two solutions to the inequality: $x < 10$ and $x > -34$. However, we need to consider the original condition $x+12 \geq 0$ and $x+12 < 0$. When $x+12 \geq 0$, the solution is $x < 10$. When $x+12 < 0$, the solution is $x > -34$. Therefore, the final solution to the inequality is $x < 10$ or $x > -34$.

Conclusion

In this article, we have solved the absolute value inequality $|x+12|+5 < 27$. We started by isolating the absolute value by subtracting 5 from both sides, and then proceeded to solve the inequality. We considered two cases: when $x+12 \geq 0$ and when $x+12 < 0$. We found that the solution to the inequality is $x < 10$ or $x > -34$.

Frequently Asked Questions

• What is the absolute value inequality? The absolute value inequality is a type of mathematical problem that involves solving equations with absolute values.
• How do I isolate the absolute value? To isolate the absolute value, you need to subtract 5 from both sides of the inequality.
• What are the two cases to consider when solving the inequality? The two cases to consider are when $x+12 \geq 0$ and when $x+12 < 0$.
• What is the final solution to the inequality? The final solution to the inequality is $x < 10$ or $x > -34$.

Step-by-Step Solution

1. Isolate the absolute value by subtracting 5 from both sides of the inequality.
2. Consider two cases: when $x+12 \geq 0$ and when $x+12 < 0$.
3. Solve the inequality for each case.
4. Combine the solutions to find the final solution to the inequality.

Example Problems

• Solve the absolute value inequality $|x-8|+3 < 15$.
• Solve the absolute value inequality $|x+5|-2 < 10$.
• Solve the absolute value inequality $|x-2|+1 < 9$.

Real-World Applications

• Absolute value inequalities are used in physics to model the motion of objects.
• Absolute value inequalities are used in engineering to design systems that can handle uncertainty.
• Absolute value inequalities are used in economics to model the behavior of markets.

Conclusion

In this article, we have solved the absolute value inequality $|x+12|+5 < 27$. We started by isolating the absolute value by subtracting 5 from both sides, and then proceeded to solve the inequality. We considered two cases: when $x+12 \geq 0$ and when $x+12 < 0$. We found that the solution to the inequality is $x < 10$ or $x > -34$.

Introduction

In our previous article, we solved the absolute value inequality $|x+12|+5 < 27$. We started by isolating the absolute value by subtracting 5 from both sides, and then proceeded to solve the inequality. In this article, we will answer some frequently asked questions about absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is a type of mathematical problem that involves solving equations with absolute values.

Q: How do I isolate the absolute value in an inequality?

A: To isolate the absolute value, you need to subtract the constant term from both sides of the inequality.

Q: What are the two cases to consider when solving an absolute value inequality?

A: The two cases to consider are when the expression inside the absolute value is positive, and when the expression inside the absolute value is negative.

Q: How do I solve an absolute value inequality when the expression inside the absolute value is positive?

A: When the expression inside the absolute value is positive, you can simply remove the absolute value signs and solve the resulting inequality.

Q: How do I solve an absolute value inequality when the expression inside the absolute value is negative?

A: When the expression inside the absolute value is negative, you need to multiply the expression by -1 and then remove the absolute value signs. You also need to flip the direction of the inequality.

Q: What is the final solution to an absolute value inequality?

A: The final solution to an absolute value inequality is the combination of the solutions to the two cases.

Q: Can I use a calculator to solve an absolute value inequality?

A: Yes, you can use a calculator to solve an absolute value inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: Can I use a graphing calculator to solve an absolute value inequality?

A: Yes, you can use a graphing calculator to solve an absolute value inequality. You can graph the two functions and find the intersection points to determine the solution.

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

A: To check your solution, you need to plug the values back into the original inequality and make sure that it is true.

Q: Can I use absolute value inequalities to model real-world problems?

A: Yes, you can use absolute value inequalities to model real-world problems. For example, you can use absolute value inequalities to model the motion of objects, the behavior of markets, and the design of systems.

Example Problems

• Solve the absolute value inequality $|x-8|+3 < 15$.
• Solve the absolute value inequality $|x+5|-2 < 10$.
• Solve the absolute value inequality $|x-2|+1 < 9$.

Real-World Applications

• Absolute value inequalities are used in physics to model the motion of objects.
• Absolute value inequalities are used in engineering to design systems that can handle uncertainty.
• Absolute value inequalities are used in economics to model the behavior of markets.

Conclusion

In this article, we have answered some frequently asked questions about absolute value inequalities. We have covered topics such as isolating the absolute value, solving the inequality, and checking the solution. We have also provided example problems and real-world applications of absolute value inequalities.

Step-by-Step Solution

1. Isolate the absolute value by subtracting the constant term from both sides of the inequality.
2. Consider two cases: when the expression inside the absolute value is positive, and when the expression inside the absolute value is negative.
3. Solve the inequality for each case.
4. Combine the solutions to find the final solution to the inequality.
5. Check the solution by plugging the values back into the original inequality.

Tips and Tricks

• Make sure to isolate the absolute value before solving the inequality.
• Consider both cases when solving the inequality.
• Check the solution by plugging the values back into the original inequality.
• Use a calculator or graphing calculator to check the solution.
• Use absolute value inequalities to model real-world problems.

Conclusion

In this article, we have provided a comprehensive guide to solving absolute value inequalities. We have covered topics such as isolating the absolute value, solving the inequality, and checking the solution. We have also provided example problems and real-world applications of absolute value inequalities.