Solve The Inequality:${ -4 + Y \leq 7 }$

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Introduction

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In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. 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 the inequality $-4 + y \leq 7$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Inequality

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The given inequality is $-4 + y \leq 7$. To solve this inequality, we need to isolate the variable $y$ on one side of the inequality sign. The inequality sign $\leq$ indicates that the value of $y$ can be equal to or greater than the value on the right-hand side of the inequality.

Step 1: Isolate the Variable

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To isolate the variable $y$, we need to get rid of the constant term $-4$ on the left-hand side of the inequality. We can do this by adding $4$ to both sides of the inequality.

-4 + y \leq 7
y + 4 \leq 7 + 4
y \leq 11

Step 2: Simplify the Inequality

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After isolating the variable $y$, we can simplify the inequality by removing the constant term from the left-hand side.

y \leq 11

Step 3: Write the Solution in Interval Notation

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The solution to the inequality $y \leq 11$ can be written in interval notation as $(-\infty, 11]$.

Conclusion

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In this article, we solved the inequality $-4 + y \leq 7$ by isolating the variable $y$ and simplifying the inequality. We also wrote the solution in interval notation as $(-\infty, 11]$. This solution indicates that the value of $y$ can be equal to or less than $11$.

Frequently Asked Questions

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Q: What is the solution to the inequality $-4 + y \leq 7$?

A: The solution to the inequality $-4 + y \leq 7$ is $y \leq 11$.

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

A: To write the solution in interval notation, you need to use the following format: $[a, b]$, where $a$ is the lower bound and $b$ is the upper bound. In this case, the solution is $(-\infty, 11]$.

Q: What is the meaning of the inequality sign $\leq$?

A: The inequality sign $\leq$ indicates that the value of $y$ can be equal to or greater than the value on the right-hand side of the inequality.

Tips and Tricks

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Tip 1: Always check your work

When solving an inequality, it's essential to check your work to ensure that you have not made any mistakes.

Tip 2: Use interval notation to write the solution

Interval notation is a convenient way to write the solution to an inequality. It helps to avoid confusion and makes it easier to understand the solution.

Tip 3: Practice, practice, practice

Solving inequalities requires practice. The more you practice, the more comfortable you will become with solving inequalities.

Final Thoughts

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Solving inequalities is an essential skill in mathematics. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to always check your work, use interval notation to write the solution, and practice regularly. With practice and patience, you will become proficient in solving inequalities.

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Introduction

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In our previous article, we solved the inequality $-4 + y \leq 7$ and wrote the solution in interval notation as $(-\infty, 11]$. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

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

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, or equal to another expression.

Q: How do I solve an inequality with a variable on both sides?

A: To solve an inequality with a variable on both sides, 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 of the inequality.

Q: What is the meaning of the inequality sign $\geq$?

A: The inequality sign $\geq$ indicates that the value of $y$ can be equal to or greater than the value on the right-hand side of the inequality.

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 use the following format: $[a, b]$, where $a$ is the lower bound and $b$ is the upper bound.

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

A: A strict inequality is an inequality that uses the inequality signs $<$ or $>$, while a non-strict inequality is an inequality that uses the inequality signs $\leq$ or $\geq$.

Q: How do I solve an inequality with a fraction?

A: To solve an inequality with a fraction, you need to get rid of the fraction by multiplying both sides of the inequality by the denominator.

Q: What is the meaning of the inequality sign $\neq$?

A: The inequality sign $\neq$ indicates that the value of $y$ is not equal to the value on the right-hand side of the inequality.

Q: How do I solve an inequality with absolute value?

A: To solve an inequality with absolute value, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

Tips and Tricks

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Tip 1: Always check your work

When solving an inequality, it's essential to check your work to ensure that you have not made any mistakes.

Tip 2: Use interval notation to write the solution

Interval notation is a convenient way to write the solution to an inequality. It helps to avoid confusion and makes it easier to understand the solution.

Tip 3: Practice, practice, practice

Solving inequalities requires practice. The more you practice, the more comfortable you will become with solving inequalities.

Real-World Applications

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Solving inequalities has many real-world applications. Here are a few examples:

• Finance: In finance, inequalities are used to calculate interest rates and investment returns.
• Science: In science, inequalities are used to model population growth and decay.
• Engineering: In engineering, inequalities are used to design and optimize systems.

Conclusion

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Solving inequalities is an essential skill in mathematics. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to always check your work, use interval notation to write the solution, and practice regularly. With practice and patience, you will become proficient in solving inequalities.

Frequently Asked Questions

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Q: What is the solution to the inequality $-4 + y \leq 7$?

A: The solution to the inequality $-4 + y \leq 7$ is $y \leq 11$.

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 use the following format: $[a, b]$, where $a$ is the lower bound and $b$ is the upper bound.

Q: What is the meaning of the inequality sign $\leq$?

A: The inequality sign $\leq$ indicates that the value of $y$ can be equal to or greater than the value on the right-hand side of the inequality.

Final Thoughts

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Solving inequalities is an essential skill in mathematics. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to always check your work, use interval notation to write the solution, and practice regularly. With practice and patience, you will become proficient in solving inequalities.