Solve The Inequality:$\[ -1 \leq 2n + 4 - 5 \\]

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Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality −1≤2n+4−5-1 \leq 2n + 4 - 5. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is −1≤2n+4−5-1 \leq 2n + 4 - 5. To solve this inequality, we need to isolate the variable nn on one side of the inequality sign. The first step is to simplify the right-hand side of the inequality by combining the constants.

Simplifying the Right-Hand Side

The right-hand side of the inequality is 2n+4−52n + 4 - 5. We can simplify this expression by combining the constants:

2n+4−5=2n−12n + 4 - 5 = 2n - 1

So, the inequality becomes:

−1≤2n−1-1 \leq 2n - 1

Adding 1 to Both Sides

To isolate the variable nn, we need to get rid of the constant term on the right-hand side. We can do this by adding 1 to both sides of the inequality:

−1+1≤2n−1+1-1 + 1 \leq 2n - 1 + 1

This simplifies to:

0≤2n0 \leq 2n

Dividing Both Sides by 2

Now that we have isolated the variable nn, we can divide both sides of the inequality by 2 to solve for nn:

02≤2n2\frac{0}{2} \leq \frac{2n}{2}

This simplifies to:

0≤n0 \leq n

Conclusion

In conclusion, the solution to the inequality −1≤2n+4−5-1 \leq 2n + 4 - 5 is 0≤n0 \leq n. This means that the value of nn must be greater than or equal to 0.

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities can be used to model the growth of investments 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.

Tips and Tricks

Here are some tips and tricks for solving inequalities:

  • Simplify the right-hand side: Before solving the inequality, simplify the right-hand side by combining constants.
  • Add or subtract the same value to both sides: To isolate the variable, add or subtract the same value to both sides of the inequality.
  • Multiply or divide both sides by the same value: To solve for the variable, multiply or divide both sides of the inequality by the same value.
  • Check your solution: After solving the inequality, check your solution by plugging it back into the original inequality.

Common Mistakes

Here are some common mistakes to avoid when solving inequalities:

  • Not simplifying the right-hand side: Failing to simplify the right-hand side can lead to incorrect solutions.
  • Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions.
  • Not multiplying or dividing both sides by the same value: Failing to multiply or divide both sides by the same value can lead to incorrect solutions.
  • Not checking your solution: Failing to check your solution can lead to incorrect solutions.

Conclusion

Introduction

In the previous article, we discussed how to solve inequalities using step-by-step examples. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to symbols.

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 to both sides, or multiplying or dividing both sides by the same value.

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

A: An equation is a mathematical expression that states that two values are equal. An inequality, on the other hand, states that two values are not equal, but one value is greater than or less than the other.

Q: How do I know which direction to move the inequality sign when adding or subtracting the same value to both sides?

A: When adding or subtracting the same value to both sides of an inequality, you need to move the inequality sign in the opposite direction. For example, if you have the inequality x>5x > 5 and you add 3 to both sides, the inequality becomes x+3>5+3x + 3 > 5 + 3, which simplifies to x>8x > 8.

Q: How do I know which direction to move the inequality sign when multiplying or dividing both sides by the same value?

A: When multiplying or dividing both sides of an inequality by a positive value, you can move the inequality sign in the same direction. However, when multiplying or dividing both sides by a negative value, you need to move the inequality sign in the opposite direction.

Q: What is the concept of "flipping" the inequality sign?

A: When multiplying or dividing both sides of an inequality by a negative value, you need to "flip" the inequality sign. This means that if you have the inequality x>5x > 5 and you multiply both sides by -1, the inequality becomes −x<−5-x < -5.

Q: How do I know when to use the "flipping" concept?

A: You need to use the "flipping" concept when multiplying or dividing both sides of an inequality by a negative value.

Q: What is the concept of "simplifying" the right-hand side of an inequality?

A: When solving an inequality, you need to simplify the right-hand side by combining constants.

Q: How do I simplify the right-hand side of an inequality?

A: To simplify the right-hand side of an inequality, you need to combine any constants on the right-hand side.

Q: What is the concept of "checking" your solution?

A: When solving an inequality, you need to check your solution by plugging it back into the original inequality.

Q: How do I check my solution?

A: To check your solution, you need to plug the solution back into the original inequality and verify that it is true.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving inequalities. Remember to simplify the right-hand side, add or subtract the same value to both sides, multiply or divide both sides by the same value, and check your solution. With this Q&A guide, you can better understand the concept of solving inequalities and become a master of solving inequalities.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving inequalities:

  • Not simplifying the right-hand side: Failing to simplify the right-hand side can lead to incorrect solutions.
  • Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions.
  • Not multiplying or dividing both sides by the same value: Failing to multiply or divide both sides by the same value can lead to incorrect solutions.
  • Not checking your solution: Failing to check your solution can lead to incorrect solutions.

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities can be used to model the growth of investments 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.

Tips and Tricks

Here are some tips and tricks for solving inequalities:

  • Simplify the right-hand side: Before solving the inequality, simplify the right-hand side by combining constants.
  • Add or subtract the same value to both sides: To isolate the variable, add or subtract the same value to both sides of the inequality.
  • Multiply or divide both sides by the same value: To solve for the variable, multiply or divide both sides of the inequality by the same value.
  • Check your solution: After solving the inequality, check your solution by plugging it back into the original inequality.