Solve The Inequality:$\[ 10x - 7 \geq 243 \\]

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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. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $10x - 7 \geq 243$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $10x - 7 \geq 243$. This means that the expression $10x - 7$ is greater than or equal to $243$. Our goal is to isolate the variable $x$ on one side of the inequality sign.

Step 1: Add 7 to Both Sides

To isolate the term $10x$, we need to get rid of the constant term $-7$ on the left side of the inequality. We can do this by adding $7$ to both sides of the inequality.

$$10x - 7 + 7 \geq 243 + 7$$

This simplifies to:

$$10x \geq 250$$

Step 2: Divide Both Sides by 10

Now that we have isolated the term $10x$, we need to get rid of the coefficient $10$ on the left side of the inequality. We can do this by dividing both sides of the inequality by $10$.

$$\frac{10x}{10} \geq \frac{250}{10}$$

This simplifies to:

$$x \geq 25$$

Conclusion

In this article, we solved the inequality $10x - 7 \geq 243$ by following a series of steps. We added $7$ to both sides of the inequality to isolate the term $10x$, and then divided both sides by $10$ to get rid of the coefficient. The final solution is $x \geq 25$. This means that any value of $x$ that is greater than or equal to $25$ will satisfy the inequality.

Tips and Tricks

• When solving inequalities, it's essential to follow the same steps as when solving equations.
• Make sure to add or subtract the same value to both sides of the inequality.
• When dividing both sides of the inequality by a negative number, flip the direction of the inequality sign.
• Always check your solution by plugging it back into the original inequality.

Real-World Applications

Solving inequalities has numerous real-world applications. For example, in finance, inequalities can be used to model investment returns and risk. In engineering, inequalities can be used to design and optimize systems. In medicine, inequalities can be used to model the spread of diseases.

Common Mistakes to Avoid

• Failing to add or subtract the same value to both sides of the inequality.
• Failing to flip the direction of the inequality sign when dividing both sides by a negative number.
• Not checking the solution by plugging it back into the original inequality.

Conclusion

Solving inequalities is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to always check your solution by plugging it back into the original inequality, and avoid common mistakes such as failing to add or subtract the same value to both sides of the inequality. With practice and patience, you can become proficient in solving inequalities and apply this skill to real-world problems.

Final Answer

The final answer is $\boxed{x \geq 25}$.

Introduction

In our previous article, we solved the inequality $10x - 7 \geq 243$ by following a series of steps. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in 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.

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. This involves adding or subtracting the same value to both sides of the inequality, and then dividing both sides by a non-zero 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 one value is greater than, less than, greater than or equal to, or less than or equal to another value.

Q: How do I know which direction to flip the inequality sign?

A: When dividing both sides of the inequality by a negative number, you need to flip the direction of the inequality sign. For example, if you have the inequality $x \geq 5$ and you divide both sides by $-2$, the inequality becomes $x \leq -\frac{5}{2}$.

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

A: Yes, you can add or subtract the same value to both sides of the inequality. However, you need to make sure that you are adding or subtracting the same value to both sides.

Q: How do I check my solution?

A: To check your solution, you need to plug it back into the original inequality. If the solution satisfies the inequality, then it is a valid solution.

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

A: Some common mistakes to avoid when solving inequalities include failing to add or subtract the same value to both sides of the inequality, failing to flip the direction of the inequality sign when dividing both sides by a negative number, and not checking the solution by plugging it back into the original inequality.

Q: How do I apply inequalities to real-world problems?

A: Inequalities can be used to model real-world problems such as investment returns and risk, system design and optimization, and disease spread. By applying inequalities to real-world problems, you can gain a deeper understanding of the underlying mathematics and make more informed decisions.

Q: What are some examples of inequalities in real-world problems?

A: Some examples of inequalities in real-world problems include:

• Investment returns: An investor may want to know the minimum return on investment required to meet a certain financial goal.
• System design: An engineer may want to design a system that meets certain performance criteria, such as speed or efficiency.
• Disease spread: A public health official may want to model the spread of a disease to predict the number of cases and develop effective interventions.

Q: How do I practice solving inequalities?

A: To practice solving inequalities, you can try solving a variety of inequalities with different coefficients and constants. You can also try applying inequalities to real-world problems to gain a deeper understanding of the underlying mathematics.

Conclusion

Solving inequalities is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article and practicing solving inequalities, you can become proficient in solving inequalities and apply this skill to real-world problems.

Final Answer

The final answer is $\boxed{x \geq 25}$.