Solve The Inequality:$\[1.25x + 3.75 \geq 22.50\\]

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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 $1.25x + 3.75 \geq 22.50$. We will use a step-by-step approach to solve this inequality and provide a clear understanding of the solution.

Understanding the Inequality

The given inequality is $1.25x + 3.75 \geq 22.50$. This inequality states that the expression $1.25x + 3.75$ is greater than or equal to $22.50$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

Step 1: Subtract 3.75 from Both Sides

The first step in solving the inequality is to subtract 3.75 from both sides of the inequality. This will help us isolate the term containing the variable $x$.

${1.25x + 3.75 - 3.75 \geq 22.50 - 3.75}$

Simplifying the inequality, we get:

${1.25x \geq 18.75}$

Step 2: Divide Both Sides by 1.25

The next step is to divide both sides of the inequality by 1.25. This will help us isolate the variable $x$.

${\frac{1.25x}{1.25} \geq \frac{18.75}{1.25}}$

Simplifying the inequality, we get:

${x \geq 15}$

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The solution $x \geq 15$ can be written as $[15, \infty)$.

Conclusion

In this article, we solved the inequality $1.25x + 3.75 \geq 22.50$ using a step-by-step approach. We subtracted 3.75 from both sides of the inequality, divided both sides by 1.25, and wrote the solution in interval notation. The solution to the inequality is $x \geq 15$, which can be written as $[15, \infty)$.

Tips and Tricks

• When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that the inequality is solved correctly.
• When subtracting or adding a value to both sides of an inequality, make sure to keep the inequality sign the same.
• When dividing both sides of an inequality by a value, make sure to check if the value is positive or negative. If the value is negative, the inequality sign will be reversed.

Common Mistakes to Avoid

• When solving inequalities, it's easy to make mistakes by forgetting to keep the inequality sign the same when subtracting or adding a value to both sides.
• When dividing both sides of an inequality by a value, make sure to check if the value is positive or negative. If the value is negative, the inequality sign will be reversed.
• When writing the solution in interval notation, make sure to include the correct interval notation.

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities are used to calculate interest rates and investment returns. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to model and analyze the spread of diseases.

Conclusion

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Introduction

In our previous article, we solved the inequality $1.25x + 3.75 \geq 22.50$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities and provide answers to common questions.

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. You can do this by adding or subtracting a value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by a value.

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: PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

Q: How do I handle negative values when solving inequalities?

A: When solving inequalities, you need to check if the value you are dividing or multiplying by is positive or negative. If the value is negative, the inequality sign will be reversed.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the inequality $x \geq 15$ can be written as $[15, \infty)$.

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 type of inequality and the value of the variable. For example, if the inequality is $x \geq 15$, the solution can be written as $[15, \infty)$.

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

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

• Forgetting to keep the inequality sign the same when subtracting or adding a value to both sides.
• Forgetting to check if the value you are dividing or multiplying by is positive or negative.
• Writing the solution to an inequality in the wrong interval notation.

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

A: Solving inequalities has many real-world applications, including finance, engineering, and medicine. For example, in finance, inequalities are used to calculate interest rates and investment returns. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to model and analyze the spread of diseases.

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

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

• Calculating interest rates and investment returns in finance.
• Designing and optimizing systems in engineering.
• Modeling and analyzing the spread of diseases in medicine.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has many real-world applications. By following a step-by-step approach and avoiding common mistakes, we can solve inequalities with confidence. In this article, we provided a Q&A guide to help you understand the concept of solving inequalities and provide answers to common questions.

Tips and Tricks

• When solving inequalities, it's essential to follow the order of operations (PEMDAS) to ensure that the inequality is solved correctly.
• When subtracting or adding a value to both sides of an inequality, make sure to keep the inequality sign the same.
• When dividing both sides of an inequality by a value, make sure to check if the value is positive or negative. If the value is negative, the inequality sign will be reversed.

Common Mistakes to Avoid

• Forgetting to keep the inequality sign the same when subtracting or adding a value to both sides.
• Forgetting to check if the value you are dividing or multiplying by is positive or negative.
• Writing the solution to an inequality in the wrong interval notation.

Real-World Applications

Solving inequalities has many real-world applications, including finance, engineering, and medicine. By applying solving inequalities to real-world problems, we can make informed decisions and solve complex problems.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has many real-world applications. By following a step-by-step approach and avoiding common mistakes, we can solve inequalities with confidence. In this article, we provided a Q&A guide to help you understand the concept of solving inequalities and provide answers to common questions.