Solve The Inequality For $v$:$-12 + V \geq -14$Simplify Your Answer As Much As Possible.
Introduction
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 linear inequalities, specifically the inequality . We will break down the solution step by step, providing a clear and concise explanation of each step.
Understanding the Inequality
The given inequality is . To solve this inequality, we need to isolate the variable on one side of the inequality sign. The inequality sign indicates that the value of can be greater than or equal to .
Step 1: Add 12 to Both Sides
To isolate the variable , we need to get rid of the constant term on the left side of the inequality. We can do this by adding to both sides of the inequality.
-12 + v \geq -14
-12 + 12 + v \geq -14 + 12
0 + v \geq -2
v \geq -2
Step 2: Simplify the Inequality
After adding to both sides, we get the simplified inequality . This means that the value of can be greater than or equal to .
Interpretation of the Solution
The solution indicates that the value of can be any real number greater than or equal to . This includes all the numbers in the interval .
Graphical Representation
To visualize the solution, we can graph the inequality on a number line. The number line represents all the possible values of . The inequality indicates that all the values of to the right of are included in the solution.
-3 -2 -1 0 1 2 3
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**Solving Inequalities: A Step-by-Step Guide**
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**Q&A: Frequently Asked Questions**
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**Q: What is an inequality?**
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A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.
**Q: How do I solve an inequality?**
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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 of the inequality.
**Q: What is the difference between a linear inequality and a quadratic inequality?**
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A: A linear inequality is an inequality that can be written in the form $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.
**Q: How do I graph an inequality on a number line?**
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A: To graph an inequality on a number line, you need to identify the values of $x$ that satisfy the inequality. If the inequality is of the form $x \geq a$, you need to shade the region to the right of $a$. If the inequality is of the form $x \leq a$, you need to shade the region to the left of $a$.
**Q: What is the solution to the inequality $-12 + v \geq -14$?**
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A: The solution to the inequality $-12 + v \geq -14$ is $v \geq -2$. This means that the value of $v$ can be any real number greater than or equal to $-2$.
**Q: How do I check my solution to an inequality?**
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A: To check your solution to an inequality, you need to plug in a value of $x$ that satisfies the inequality and verify that the inequality is true. If the inequality is true, then your solution is correct.
**Q: What are some common mistakes to avoid when solving inequalities?**
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A: Some common mistakes to avoid when solving inequalities include:
* Adding or subtracting the wrong value to both sides of the inequality
* Multiplying or dividing both sides of the inequality by a negative number
* Not checking the solution to the inequality
**Q: How do I use inequalities in real-life situations?**
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A: Inequalities are used in a wide range of real-life situations, including:
* Finance: Inequalities are used to compare the value of different investments.
* Science: Inequalities are used to compare the values of different physical quantities.
* Engineering: Inequalities are used to compare the values of different engineering quantities.
**Conclusion**
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Solving inequalities is an important skill that is used in a wide range of real-life situations. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to check your solution to an inequality and to avoid common mistakes when solving inequalities.
**Additional Resources**
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For more information on solving inequalities, check out the following resources:
* Khan Academy: Solving Inequalities
* Mathway: Solving Inequalities
* Wolfram Alpha: Solving Inequalities
**Practice Problems**
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Try solving the following inequalities:
* $2x + 3 \geq 5$
* $x - 2 \leq 3$
* $x^2 + 4x + 4 \geq 0$
**Answer Key**
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* $x \geq 1$
* $x \leq 5$
* $(x + 2)^2 \geq 0$