What Are The Two Equations Created By The Inequality $|x-12|+5\ \textless \ 27$?$y_1 = $ $\square$y_2 = $

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Introduction

In mathematics, inequalities are used to describe the relationship between two or more expressions. The absolute value inequality is a type of inequality that involves the absolute value of an expression. In this article, we will explore the two equations created by the inequality $|x-12|+5\ \textless \ 27$. We will start by understanding the concept of absolute value and then proceed to solve the inequality.

Understanding Absolute Value

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It is denoted by the symbol $|x|$. The absolute value of a number is always non-negative, i.e., it is always greater than or equal to zero. For example, the absolute value of $-3$ is $3$, and the absolute value of $5$ is $5$.

Solving the Inequality

To solve the inequality $|x-12|+5\ \textless \ 27$, we need to isolate the absolute value expression. We can do this by subtracting $5$ from both sides of the inequality, which gives us $|x-12|\ \textless \ 22$.

Case 1: $x-12 \geq 0$

When $x-12 \geq 0$, the absolute value expression $|x-12|$ simplifies to $x-12$. Therefore, the inequality becomes $x-12\ \textless \ 22$. We can add $12$ to both sides of the inequality to get $x\ \textless \ 34$.

Case 2: $x-12 < 0$

When $x-12 < 0$, the absolute value expression $|x-12|$ simplifies to $-(x-12)$. Therefore, the inequality becomes $-(x-12)\ \textless \ 22$. We can simplify this inequality by multiplying both sides by $-1$, which gives us $x-12\ \textgreater \ -22$. We can add $12$ to both sides of the inequality to get $x\ \textgreater \ -10$.

Conclusion

In conclusion, the two equations created by the inequality $|x-12|+5\ \textless \ 27$ are $y_1 = x\ \textless \ 34$ and $y_2 = x\ \textgreater \ -10$. These equations represent the solution set of the inequality, which is the set of all values of $x$ that satisfy the inequality.

Graphical Representation

The solution set of the inequality can be represented graphically on a number line. The number line is divided into two parts: one part represents the values of $x$ that satisfy the inequality $x\ \textless \ 34$, and the other part represents the values of $x$ that satisfy the inequality $x\ \textgreater \ -10$.

Final Answer

The final answer is $\boxed{y_1 = x\ \textless \ 34, y_2 = x\ \textgreater \ -10}$.

Frequently Asked Questions

• What is the absolute value inequality?
• How do you solve an absolute value inequality?
• What are the two equations created by the inequality $|x-12|+5\ \textless \ 27$?

Answer to Frequently Asked Questions

• The absolute value inequality is a type of inequality that involves the absolute value of an expression.
• To solve an absolute value inequality, you need to isolate the absolute value expression and then consider two cases: one case when the expression is non-negative and another case when the expression is negative.
• The two equations created by the inequality $|x-12|+5\ \textless \ 27$ are $y_1 = x\ \textless \ 34$ and $y_2 = x\ \textgreater \ -10$.

References

• [1] "Absolute Value Inequalities" by Math Open Reference
• [2] "Solving Absolute Value Inequalities" by Purplemath
• [3] "Absolute Value Inequalities" by Khan Academy
  

Introduction

In our previous article, we explored the concept of absolute value inequalities and solved the inequality $|x-12|+5\ \textless \ 27$. In this article, we will answer some frequently asked questions related to absolute value inequalities.

Q&A

Q1: What is the absolute value inequality?

A1: The absolute value inequality is a type of 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.

Q2: How do you solve an absolute value inequality?

A2: To solve an absolute value inequality, you need to isolate the absolute value expression and then consider two cases: one case when the expression is non-negative and another case when the expression is negative.

Q3: What are the two equations created by the inequality $|x-12|+5\ \textless \ 27$?

A3: The two equations created by the inequality $|x-12|+5\ \textless \ 27$ are $y_1 = x\ \textless \ 34$ and $y_2 = x\ \textgreater \ -10$.

Q4: How do you graph the solution set of an absolute value inequality?

A4: To graph the solution set of an absolute value inequality, you need to draw a number line and mark the values of $x$ that satisfy the inequality. You can use a closed circle to represent the values of $x$ that satisfy the inequality, and an open circle to represent the values of $x$ that do not satisfy the inequality.

Q5: Can you give an example of an absolute value inequality?

A5: Yes, here is an example of an absolute value inequality: $|x+3|\ \textless \ 5$. To solve this inequality, you need to isolate the absolute value expression and then consider two cases: one case when the expression is non-negative and another case when the expression is negative.

Q6: How do you solve an absolute value inequality with a negative number inside the absolute value?

A6: To solve an absolute value inequality with a negative number inside the absolute value, you need to multiply the inequality by $-1$ and then change the direction of the inequality sign.

Q7: Can you give an example of an absolute value inequality with a negative number inside the absolute value?

A7: Yes, here is an example of an absolute value inequality with a negative number inside the absolute value: $|x-5|\ \textless \ 2$. To solve this inequality, you need to multiply the inequality by $-1$ and then change the direction of the inequality sign.

Q8: How do you solve an absolute value inequality with a fraction inside the absolute value?

A8: To solve an absolute value inequality with a fraction inside the absolute value, you need to multiply the inequality by the reciprocal of the fraction and then simplify the expression.

Q9: Can you give an example of an absolute value inequality with a fraction inside the absolute value?

A9: Yes, here is an example of an absolute value inequality with a fraction inside the absolute value: $|\frac{x}{2}|\ \textless \ 3$. To solve this inequality, you need to multiply the inequality by $\frac{2}{1}$ and then simplify the expression.

Q10: How do you solve an absolute value inequality with a variable inside the absolute value?

A10: To solve an absolute value inequality with a variable inside the absolute value, you need to isolate the absolute value expression and then consider two cases: one case when the expression is non-negative and another case when the expression is negative.

Conclusion

In conclusion, absolute value inequalities are a type of inequality that involves the absolute value of an expression. To solve an absolute value inequality, you need to isolate the absolute value expression and then consider two cases: one case when the expression is non-negative and another case when the expression is negative. We hope that this article has helped you to understand absolute value inequalities and how to solve them.

Final Answer

The final answer is $\boxed{y_1 = x\ \textless \ 34, y_2 = x\ \textgreater \ -10}$.

References

• [1] "Absolute Value Inequalities" by Math Open Reference
• [2] "Solving Absolute Value Inequalities" by Purplemath
• [3] "Absolute Value Inequalities" by Khan Academy