Solve The Inequality: $-4 \leq 2(x+1) \ \textless \ 10$

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Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, specifically the inequality −4≤2(x+1) \textless 10-4 \leq 2(x+1) \ \textless \ 10. We will break down the solution process into manageable steps, making it easier for readers to understand and apply the concepts.

Understanding the Inequality

The given inequality is −4≤2(x+1) \textless 10-4 \leq 2(x+1) \ \textless \ 10. This is a compound inequality, which means it consists of two parts: a less-than-or-equal-to part and a greater-than part. Our goal is to solve for the variable xx.

Distributing the Coefficient

To begin solving the inequality, we need to distribute the coefficient 22 to the terms inside the parentheses. This will help us simplify the expression and make it easier to work with.

2(x+1) = 2x + 2

Now, the inequality becomes −4≤2x+2 \textless 10-4 \leq 2x + 2 \ \textless \ 10.

Subtracting 2 from Both Sides

Next, we need to isolate the term with the variable xx. To do this, we subtract 22 from both sides of the inequality.

-4 - 2 \leq 2x + 2 - 2 \  \textless \  10 - 2

This simplifies to −6≤2x \textless 8-6 \leq 2x \ \textless \ 8.

Dividing Both Sides by 2

Now, we need to divide both sides of the inequality by 22 to solve for xx.

\frac{-6}{2} \leq \frac{2x}{2} \  \textless \  \frac{8}{2}

This simplifies to −3≤x \textless 4-3 \leq x \ \textless \ 4.

Conclusion

In this article, we solved the inequality −4≤2(x+1) \textless 10-4 \leq 2(x+1) \ \textless \ 10 using a step-by-step approach. We distributed the coefficient, subtracted 22 from both sides, and finally divided both sides by 22 to solve for xx. The solution to the inequality is −3≤x \textless 4-3 \leq x \ \textless \ 4. This means that the value of xx must be greater than or equal to −3-3 and less than 44.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

  • Always distribute the coefficient to the terms inside the parentheses.
  • Subtract or add the same value to both sides of the inequality.
  • Divide both sides of the inequality by the same value.
  • Be careful when multiplying or dividing both sides of the inequality by a negative value, as it will change the direction of the inequality.

Practice Problems

To reinforce your understanding of solving inequalities, try the following practice problems:

  • Solve the inequality 3≤2x−2 \textless 73 \leq 2x - 2 \ \textless \ 7.
  • Solve the inequality −5≤x+3 \textless 2-5 \leq x + 3 \ \textless \ 2.
  • Solve the inequality 4≤2x+1 \textless 94 \leq 2x + 1 \ \textless \ 9.

Final Thoughts

Solving inequalities is a crucial skill in mathematics, and with practice, you can become proficient in solving various types of inequalities. Remember to always follow the steps outlined in this article, and don't hesitate to ask for help if you need it. With patience and persistence, you will become a master of solving inequalities in no time.

References

Related Articles

FAQs

  • Q: What is an inequality?
  • A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
  • Q: How do I solve an inequality?
  • A: To solve an inequality, follow the steps outlined in this article, including distributing the coefficient, subtracting or adding the same value to both sides, and dividing both sides by the same value.
  • Q: What are some common mistakes to avoid when solving inequalities?
  • A: Some common mistakes to avoid when solving inequalities include forgetting to distribute the coefficient, subtracting or adding the wrong value to both sides, and dividing both sides by a negative value without changing the direction of the inequality.

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Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. Inequalities are used to describe relationships between variables and are a fundamental concept in mathematics.

Q: How do I solve an inequality?

A: To solve an inequality, follow these steps:

  1. Distribute the coefficient to the terms inside the parentheses.
  2. Subtract or add the same value to both sides of the inequality.
  3. Divide both sides of the inequality by the same value.
  4. Be careful when multiplying or dividing both sides of the inequality by a negative value, as it will change the direction of the inequality.

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

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

  • Forgetting to distribute the coefficient to the terms inside the parentheses.
  • Subtracting or adding the wrong value to both sides of the inequality.
  • Dividing both sides of the inequality by a negative value without changing the direction of the inequality.
  • Not considering the direction of the inequality when multiplying or dividing both sides.

Q: How do I determine the direction of the inequality?

A: The direction of the inequality is determined by the sign of the coefficient. If the coefficient is positive, the inequality remains the same direction. If the coefficient is negative, the inequality changes direction.

Q: Can I use the same steps to solve quadratic inequalities?

A: Yes, the same steps can be used to solve quadratic inequalities. However, quadratic inequalities may require additional steps, such as factoring or using the quadratic formula.

Q: How do I graph an inequality?

A: To graph an inequality, follow these steps:

  1. Plot the boundary line of the inequality.
  2. Test a point on either side of the boundary line to determine the direction of the inequality.
  3. Shade the region that satisfies the inequality.

Q: Can I use a calculator to solve inequalities?

A: Yes, many calculators have built-in functions for solving inequalities. However, it's essential to understand the steps involved in solving inequalities to ensure accurate results.

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

A: To check your solution to an inequality, substitute the value of the variable into the original inequality and verify that it is true.

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

A: Inequalities have numerous real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates and investment returns.
  • Science: Inequalities are used to model population growth and decay.
  • Engineering: Inequalities are used to design and optimize systems.

Additional Resources

Conclusion

Solving inequalities is a crucial skill in mathematics, and with practice, you can become proficient in solving various types of inequalities. Remember to always follow the steps outlined in this article, and don't hesitate to ask for help if you need it. With patience and persistence, you will become a master of solving inequalities in no time.

Final Thoughts

Inequalities are a fundamental concept in mathematics, and understanding how to solve them is essential for success in various fields. By following the steps outlined in this article and practicing regularly, you will become proficient in solving inequalities and be able to apply this skill to real-world problems.

References

Related Articles

FAQs

  • Q: What is an inequality?
  • A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
  • Q: How do I solve an inequality?
  • A: To solve an inequality, follow the steps outlined in this article, including distributing the coefficient, subtracting or adding the same value to both sides, and dividing both sides by the same value.
  • Q: What are some common mistakes to avoid when solving inequalities?
  • A: Some common mistakes to avoid when solving inequalities include forgetting to distribute the coefficient, subtracting or adding the wrong value to both sides, and dividing both sides by a negative value without changing the direction of the inequality.