Solve The Inequality $6 - 2x + 1 \ \textless \ -3(5 - X) + 7$.

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. Inequalities are mathematical expressions that compare two values, and they can be used to model real-world problems. Solving an inequality involves finding the values of the variable that make the inequality true. In this case, we will solve the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$.

Understanding the Inequality

Before we start solving the inequality, let's break it down and understand what it means. The inequality is given as $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$. This can be rewritten as $7 - 2x \ \textless \ -15 + 3x + 7$. Simplifying further, we get $7 - 2x \ \textless \ -8 + 3x$.

Isolating the Variable

To solve the inequality, we need to isolate the variable x. We can do this by adding 2x to both sides of the inequality, which gives us $7 \ \textless \ -8 + 5x$. Next, we can add 8 to both sides of the inequality, which gives us $15 \ \textless \ 5x$.

Solving for x

Now that we have isolated the variable x, we can solve for x. To do this, we can divide both sides of the inequality by 5, which gives us $3 \ \textless \ x$. This means that x must be greater than 3.

Writing the Solution in Interval Notation

The solution to the inequality can be written in interval notation as $(3, \infty)$. This means that x can take on any value greater than 3.

Conclusion

In this article, we learned how to solve the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$. We broke down the inequality, isolated the variable x, and solved for x. The solution to the inequality is x > 3, which can be written in interval notation as $(3, \infty)$.

Tips and Tricks

  • When solving an inequality, it's essential to follow the order of operations (PEMDAS) and to isolate the variable.
  • When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
  • When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.

Frequently Asked Questions

  • Q: What is the solution to the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$? A: The solution to the inequality is x > 3, which can be written in interval notation as $(3, \infty)$.
  • Q: How do I solve an inequality? A: To solve an inequality, you need to isolate the variable and follow the order of operations (PEMDAS).
  • Q: What is the difference between an inequality and an equation? A: An inequality is a mathematical expression that compares two values, while an equation is a mathematical expression that states that two values are equal.

Real-World Applications

Inequalities have many real-world applications, including:

  • Modeling population growth
  • Calculating interest rates
  • Determining the maximum or minimum value of a function
  • Solving optimization problems

Conclusion

In this article, we learned how to solve the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$. We broke down the inequality, isolated the variable x, and solved for x. The solution to the inequality is x > 3, which can be written in interval notation as $(3, \infty)$. Inequalities have many real-world applications, and solving them is an essential skill in mathematics and science.

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Introduction

In our previous article, we learned how to solve the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$. In this article, we will answer some frequently asked questions about inequalities and provide additional tips and tricks for solving them.

Q&A

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

A: An inequality is a mathematical expression that compares two values, while an equation is a mathematical expression that states that two values are equal.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable and follow the order of operations (PEMDAS). You can add or subtract the same value to both sides of the inequality, but you need to reverse the direction of the inequality when multiplying or dividing both sides by a negative value.

Q: What is the solution to the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$?

A: The solution to the inequality is x > 3, which can be written in interval notation as $(3, \infty)$.

Q: How do I write the solution to an inequality in interval notation?

A: To write the solution to an inequality in interval notation, you need to determine the values of the variable that make the inequality true. For example, if the inequality is x > 3, the solution in interval notation is $(3, \infty)$.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as < or >. A non-strict inequality is an inequality that is written with a non-strict symbol, such as ≤ or ≥.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values of the variable that make all of the inequalities true. You can use the same methods as solving a single inequality, but you need to consider all of the inequalities in the system.

Q: What is the importance of inequalities in real-world applications?

A: Inequalities have many real-world applications, including modeling population growth, calculating interest rates, determining the maximum or minimum value of a function, and solving optimization problems.

Tips and Tricks

  • When solving an inequality, it's essential to follow the order of operations (PEMDAS) and to isolate the variable.
  • When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
  • When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.
  • When solving a system of inequalities, you need to consider all of the inequalities in the system.
  • When writing the solution to an inequality in interval notation, you need to determine the values of the variable that make the inequality true.

Real-World Applications

Inequalities have many real-world applications, including:

  • Modeling population growth
  • Calculating interest rates
  • Determining the maximum or minimum value of a function
  • Solving optimization problems
  • Determining the range of a function

Conclusion

In this article, we answered some frequently asked questions about inequalities and provided additional tips and tricks for solving them. Inequalities have many real-world applications, and solving them is an essential skill in mathematics and science.

Additional Resources

  • Khan Academy: Inequalities
  • Mathway: Inequalities
  • Wolfram Alpha: Inequalities

Conclusion

In this article, we learned how to solve the inequality $6 - 2x + 1 \ \textless \ -3(5 - x) + 7$ and answered some frequently asked questions about inequalities. We also provided additional tips and tricks for solving them. Inequalities have many real-world applications, and solving them is an essential skill in mathematics and science.