Solve X + 2 1 2 \textless 8 X + 2\frac{1}{2} \ \textless \ 8 X + 2 2 1 ​ \textless 8 . Make Sure To Write Your Inequality So That X X X Comes First.

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Introduction

In this article, we will be solving the inequality $x + 2\frac{1}{2} \ \textless \ 8$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. We will use basic algebraic operations to simplify the inequality and find the solution.

Understanding the Inequality

The given inequality is $x + 2\frac{1}{2} \ \textless \ 8$. To start solving this inequality, we need to understand the concept of inequalities and how to work with them. An inequality is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other. In this case, we have a less-than inequality, which means that the value of $x$ plus $2\frac{1}{2}$ is less than $8$.

Simplifying the Inequality

To simplify the inequality, we need to get rid of the fraction $2\frac{1}{2}$. We can do this by converting the mixed number to an improper fraction. The mixed number $2\frac{1}{2}$ can be written as $\frac{5}{2}$. Now, we can rewrite the inequality as $x + \frac{5}{2} \ \textless \ 8$.

Isolating the Variable

To isolate the variable $x$, we need to get rid of the fraction $\frac{5}{2}$. We can do this by subtracting $\frac{5}{2}$ from both sides of the inequality. This will give us $x \ \textless \ 8 - \frac{5}{2}$.

Simplifying the Right-Hand Side

To simplify the right-hand side of the inequality, we need to subtract $\frac{5}{2}$ from $8$. We can do this by finding a common denominator, which is $2$. Then, we can rewrite $8$ as $\frac{16}{2}$. Now, we can subtract $\frac{5}{2}$ from $\frac{16}{2}$ to get $\frac{11}{2}$.

Writing the Final Inequality

Now that we have isolated the variable $x$, we can write the final inequality as $x \ \textless \ \frac{11}{2}$. This means that the value of $x$ is less than $\frac{11}{2}$.

Conclusion

In this article, we solved the inequality $x + 2\frac{1}{2} \ \textless \ 8$ by isolating the variable $x$ on one side of the inequality sign. We simplified the inequality by converting the mixed number to an improper fraction and then subtracting the fraction from both sides. Finally, we wrote the final inequality as $x \ \textless \ \frac{11}{2}$.

Example

Let's consider an example to illustrate the concept. Suppose we have the inequality $x + 3 \ \textless \ 10$. To solve this inequality, we can follow the same steps as before. First, we can simplify the inequality by converting the mixed number to an improper fraction. Then, we can isolate the variable $x$ by subtracting the fraction from both sides. Finally, we can write the final inequality as $x \ \textless \ 7$.

Graphing the Inequality

To visualize the inequality, we can graph it on a number line. The inequality $x \ \textless \ \frac{11}{2}$ means that the value of $x$ is less than $\frac{11}{2}$. We can represent this on a number line by drawing a point at $\frac{11}{2}$ and shading the region to the left of the point.

Real-World Applications

Inequalities have many real-world applications. For example, in finance, inequalities can be used to model the growth of an investment over time. In engineering, inequalities can be used to design and optimize systems. In medicine, inequalities can be used to model the spread of diseases.

Final Thoughts

In this article, we solved the inequality $x + 2\frac{1}{2} \ \textless \ 8$ by isolating the variable $x$ on one side of the inequality sign. We simplified the inequality by converting the mixed number to an improper fraction and then subtracting the fraction from both sides. Finally, we wrote the final inequality as $x \ \textless \ \frac{11}{2}$. We also discussed the importance of inequalities in real-world applications and provided an example to illustrate the concept.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Inequalities: Theory, Applications, and Algorithms" by László Lovász

Further Reading

• [1] "Solving Inequalities" by Khan Academy
• [2] "Inequalities" by Math Open Reference
• [3] "Algebra and Inequalities" by MIT OpenCourseWare
  

Introduction

In our previous article, we solved the inequality $x + 2\frac{1}{2} \ \textless \ 8$ by isolating the variable $x$ on one side of the inequality sign. We simplified the inequality by converting the mixed number to an improper fraction and then subtracting the fraction from both sides. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, or equal to another expression.

Q: Can I add or subtract the same value from both sides of an inequality?

A: Yes, you can add or subtract the same value from both sides of an inequality. For example, if you have the inequality $x + 2 \ \textless \ 5$, you can subtract 2 from both sides to get $x \ \textless \ 3$.

Q: Can I multiply or divide both sides of an inequality by the same non-zero value?

A: Yes, you can multiply or divide both sides of an inequality by the same non-zero value. For example, if you have the inequality $x \ \textless \ 4$, you can multiply both sides by 2 to get $2x \ \textless \ 8$.

Q: What is the order of operations for solving inequalities?

A: The order of operations for solving inequalities is the same as for solving equations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: Can I use inverse operations to solve an inequality?

A: Yes, you can use inverse operations to solve an inequality. For example, if you have the inequality $x + 3 \ \textless \ 7$, you can subtract 3 from both sides to get $x \ \textless \ 4$.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to draw a point at the value of the expression on the right-hand side of the inequality, and then shade the region to the left or right of the point, depending on whether the inequality is less than or greater than.

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

A: Yes, you can use inequalities to model real-world problems. For example, you can use inequalities to model the growth of an investment over time, or to design and optimize systems.

Q: What are some common types of inequalities?

A: Some common types of inequalities include linear inequalities, quadratic inequalities, and rational inequalities.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the factored form to determine the solution set.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to factor the numerator and denominator of the rational expression and then use the factored form to determine the solution set.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We discussed the basics of inequalities, including how to solve them and how to graph them on a number line. We also discussed some common types of inequalities and how to solve them.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Inequalities: Theory, Applications, and Algorithms" by László Lovász

Further Reading

• [1] "Solving Inequalities" by Khan Academy
• [2] "Inequalities" by Math Open Reference
• [3] "Algebra and Inequalities" by MIT OpenCourseWare