Solve The Inequality:$\[ 36 + 12m \ \textgreater \ 0 \\]
Introduction
In this article, we will delve into the world of inequalities and focus on solving the given inequality: 36 + 12m > 0. Inequalities are mathematical expressions that contain variables and constants, and they are used to describe relationships between quantities. Solving an inequality involves finding the values of the variable that satisfy the given inequality. In this case, we will use algebraic techniques to solve the inequality and find the range of values for m that satisfy the given inequality.
Understanding the Inequality
The given inequality is 36 + 12m > 0. This inequality states that the expression 36 + 12m is greater than zero. To solve this inequality, we need to isolate the variable m. We can start by subtracting 36 from both sides of the inequality, which gives us:
12m > -36
Isolating the Variable
To isolate the variable m, we need to get rid of the coefficient 12 that is being multiplied by m. We can do this by dividing both sides of the inequality by 12. However, we need to be careful when dividing by a negative number, as it will change the direction of the inequality. In this case, we are dividing by 12, which is a positive number, so the direction of the inequality will remain the same.
m > -3
Analyzing the Solution
The solution to the inequality is m > -3. This means that any value of m that is greater than -3 will satisfy the given inequality. In other words, the range of values for m that satisfy the inequality is all real numbers greater than -3.
Graphical Representation
To visualize the solution, we can graph the inequality on a number line. We can draw a vertical line at x = -3, and shade the region to the right of the line. This represents all the values of m that satisfy the inequality.
Conclusion
In this article, we solved the inequality 36 + 12m > 0 using algebraic techniques. We started by subtracting 36 from both sides of the inequality, and then divided both sides by 12 to isolate the variable m. The solution to the inequality is m > -3, which represents all real numbers greater than -3. We also graphed the inequality on a number line to visualize the solution.
Frequently Asked Questions
- What is the solution to the inequality 36 + 12m > 0?
- How do you solve an inequality?
- What is the range of values for m that satisfy the inequality 36 + 12m > 0?
Answer
- The solution to the inequality 36 + 12m > 0 is m > -3.
- To solve an inequality, you need to isolate the variable by performing algebraic operations such as addition, subtraction, multiplication, and division.
- The range of values for m that satisfy the inequality 36 + 12m > 0 is all real numbers greater than -3.
Related Topics
- Solving linear inequalities
- Graphing inequalities on a number line
- Algebraic techniques for solving inequalities
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
- [3] "Mathematics for Calculus" by Michael Sullivan
Further Reading
- For more information on solving inequalities, please refer to the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Wolfram Alpha: Solving Inequalities
Introduction
In our previous article, we solved the inequality 36 + 12m > 0 using algebraic techniques. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving inequalities. Whether you are a student, teacher, or simply someone who wants to learn more about inequalities, this guide is for you.
Q&A
Q1: What is an inequality?
A1: An inequality is a mathematical expression that contains variables and constants, and it is used to describe relationships between quantities. Inequalities can be either greater than (>) or less than (<), greater than or equal to (≥), or less than or equal to (≤).
Q2: How do I solve an inequality?
A2: To solve an inequality, you need to isolate the variable by performing algebraic operations such as addition, subtraction, multiplication, and division. You can also use inverse operations to get rid of the coefficient of the variable.
Q3: What is the difference between a linear inequality and a quadratic inequality?
A3: A linear inequality is an inequality that can be written in the form ax + b > c or ax + b < c, where a, b, and c are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax^2 + bx + c > d or ax^2 + bx + c < d, where a, b, c, and d are constants.
Q4: How do I graph an inequality on a number line?
A4: To graph an inequality on a number line, you need to draw a vertical line at the value of the variable that makes the inequality true. If the inequality is greater than or equal to, you need to shade the region to the right of the line. If the inequality is less than or equal to, you need to shade the region to the left of the line.
Q5: What is the solution to the inequality 2x - 5 > 3?
A5: To solve the inequality 2x - 5 > 3, you need to add 5 to both sides of the inequality, which gives you 2x > 8. Then, you need to divide both sides of the inequality by 2, which gives you x > 4.
Q6: How do I solve an inequality with fractions?
A6: To solve an inequality with fractions, you need to get rid of the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.
Q7: What is the range of values for x that satisfy the inequality x^2 + 4x + 4 > 0?
A7: To solve the inequality x^2 + 4x + 4 > 0, you need to factor the left-hand side of the inequality, which gives you (x + 2)^2 > 0. Since the square of any real number is always non-negative, the inequality is true for all real values of x.
Q8: How do I solve an inequality with absolute value?
A8: To solve an inequality with absolute value, you need to consider two cases: when the expression inside the absolute value is positive, and when it is negative. You can then use the properties of absolute value to simplify the inequality.
Q9: What is the solution to the inequality |x - 2| > 3?
A9: To solve the inequality |x - 2| > 3, you need to consider two cases: when x - 2 > 3, and when x - 2 < -3. You can then use the properties of absolute value to simplify the inequality and find the solution.
Q10: How do I check my solution to an inequality?
A10: To check your solution to an inequality, you need to plug in a value of the variable that satisfies the inequality and make sure that the inequality is true. You can also use a number line to visualize the solution and check if it is correct.
Conclusion
In this Q&A guide, we have provided answers to some of the most common questions about solving inequalities. Whether you are a student, teacher, or simply someone who wants to learn more about inequalities, we hope that this guide has been helpful. Remember to practice solving inequalities regularly to become more confident and proficient in solving them.
Frequently Asked Questions
- What is an inequality?
- How do I solve an inequality?
- What is the difference between a linear inequality and a quadratic inequality?
- How do I graph an inequality on a number line?
- What is the solution to the inequality 2x - 5 > 3?
Answer
- An inequality is a mathematical expression that contains variables and constants, and it is used to describe relationships between quantities.
- To solve an inequality, you need to isolate the variable by performing algebraic operations such as addition, subtraction, multiplication, and division.
- A linear inequality is an inequality that can be written in the form ax + b > c or ax + b < c, where a, b, and c are constants.
- To graph an inequality on a number line, you need to draw a vertical line at the value of the variable that makes the inequality true.
- The solution to the inequality 2x - 5 > 3 is x > 4.
Related Topics
- Solving linear inequalities
- Graphing inequalities on a number line
- Algebraic techniques for solving inequalities
- Absolute value inequalities
- Quadratic inequalities
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
- [3] "Mathematics for Calculus" by Michael Sullivan
Further Reading
- For more information on solving inequalities, please refer to the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Wolfram Alpha: Solving Inequalities