Solve The Inequality:${ \left|\frac{1}{2} X - 45\right| \geq 80 }$

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Absolute value inequalities are a specific type of inequality that involves the absolute value of an expression. In this article, we will focus on solving the inequality $\left|\frac{1}{2} x - 45\right| \geq 80$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of an expression, which is denoted by the symbol $|x|$. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

When solving absolute value inequalities, we need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

Case 1: $\frac{1}{2} x - 45 \geq 80$

In this case, we can start by adding $45$ to both sides of the inequality, which gives us $\frac{1}{2} x \geq 125$. To isolate $x$, we can multiply both sides of the inequality by $2$, which gives us $x \geq 250$.

Case 2: $\frac{1}{2} x - 45 \leq -80$

In this case, we can start by adding $45$ to both sides of the inequality, which gives us $\frac{1}{2} x \leq -35$. To isolate $x$, we can multiply both sides of the inequality by $2$, which gives us $x \leq -70$.

Combining the Cases

Now that we have solved the two cases, we can combine them to get the final solution. We know that $x \geq 250$ or $x \leq -70$. This can be written as a compound inequality: $x \in [-70, 250]$.

Graphing the Solution

To visualize the solution, we can graph the two lines $x = 250$ and $x = -70$ on a number line. The solution is all the points on the number line that are to the left of $x = -70$ or to the right of $x = 250$.

Conclusion

Solving absolute value inequalities requires us to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. By following the steps outlined in this article, we can solve the inequality $\left|\frac{1}{2} x - 45\right| \geq 80$ and find the solution $x \in [-70, 250]$.

Additional Tips and Tricks

• When solving absolute value inequalities, make sure to consider both cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.
• Use a number line to visualize the solution and make sure to include all the points that satisfy the inequality.
• When combining the cases, make sure to use the correct notation, such as $x \in [-70, 250]$.

Frequently Asked Questions

• Q: What is the solution to the inequality $\left|\frac{1}{2} x - 45\right| \geq 80$? A: The solution is $x \in [-70, 250]$.
• Q: How do I solve absolute value inequalities? A: To solve absolute value inequalities, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.
• Q: What is the difference between an absolute value inequality and a regular inequality? A: An absolute value inequality involves the absolute value of an expression, while a regular inequality does not.

Final Thoughts

Solving absolute value inequalities requires a clear understanding of the concept of absolute value and how to apply it to inequalities. By following the steps outlined in this article, you can solve absolute value inequalities and find the solution. Remember to consider both cases and use a number line to visualize the solution. With practice and patience, you will become proficient in solving absolute value inequalities.

Introduction

In our previous article, we discussed how to solve absolute value inequalities. However, we know that there are many more questions and concerns that students and educators may have. In this article, we will address some of the most frequently asked questions about absolute value inequalities.

Q&A

Q: What is the definition of an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of an expression. It is denoted by the symbol $|x|$, where $x$ is the expression inside the absolute value.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

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

A: An absolute value inequality involves the absolute value of an expression, while a regular inequality does not. For example, the inequality $|x| \geq 2$ is an absolute value inequality, while the inequality $x \geq 2$ is a regular inequality.

Q: How do I graph an absolute value inequality?

A: To graph an absolute value inequality, you need to graph the two lines that are the boundary of the inequality. For example, if the inequality is $|x| \geq 2$, you would graph the two lines $x = 2$ and $x = -2$.

Q: What is the solution to the inequality $|x| \geq 2$?

A: The solution to the inequality $|x| \geq 2$ is $x \in [-\infty, -2] \cup [2, \infty]$.

Q: How do I solve an absolute value inequality with fractions?

A: To solve an absolute value inequality with fractions, you need to follow the same steps as solving an absolute value inequality with integers. However, you may need to multiply both sides of the inequality by the denominator to eliminate the fraction.

Q: What is the solution to the inequality $|x/2| \geq 3$?

A: The solution to the inequality $|x/2| \geq 3$ is $x \in [-6, 6]$.

Q: How do I solve an absolute value inequality with decimals?

A: To solve an absolute value inequality with decimals, you need to follow the same steps as solving an absolute value inequality with integers. However, you may need to multiply both sides of the inequality by the denominator to eliminate the decimal.

Q: What is the solution to the inequality $|x/0.5| \geq 2.5$?

A: The solution to the inequality $|x/0.5| \geq 2.5$ is $x \in [-1.25, 1.25]$.

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

A: To check your solution to an absolute value inequality, you need to plug in a value from the solution set into the original inequality and make sure that it is true.

Conclusion

Solving absolute value inequalities can be challenging, but with practice and patience, you can become proficient in solving them. Remember to consider both cases and use a number line to visualize the solution. If you have any further questions or concerns, feel free to ask.

Additional Tips and Tricks

• When solving absolute value inequalities, make sure to consider both cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.
• Use a number line to visualize the solution and make sure to include all the points that satisfy the inequality.
• When combining the cases, make sure to use the correct notation, such as $x \in [-70, 250]$.
• When solving absolute value inequalities with fractions or decimals, make sure to multiply both sides of the inequality by the denominator to eliminate the fraction or decimal.
• When checking your solution, make sure to plug in a value from the solution set into the original inequality and make sure that it is true.

Frequently Asked Questions

• Q: What is the solution to the inequality $|x| \geq 2$? A: The solution to the inequality $|x| \geq 2$ is $x \in [-\infty, -2] \cup [2, \infty]$.
• Q: How do I solve an absolute value inequality with fractions? A: To solve an absolute value inequality with fractions, you need to follow the same steps as solving an absolute value inequality with integers. However, you may need to multiply both sides of the inequality by the denominator to eliminate the fraction.
• Q: What is the solution to the inequality $|x/2| \geq 3$? A: The solution to the inequality $|x/2| \geq 3$ is $x \in [-6, 6]$.

Final Thoughts

Solving absolute value inequalities requires a clear understanding of the concept of absolute value and how to apply it to inequalities. By following the steps outlined in this article, you can solve absolute value inequalities and find the solution. Remember to consider both cases and use a number line to visualize the solution. With practice and patience, you will become proficient in solving absolute value inequalities.