Solve The Inequality: ${ 9 - \frac{x}{4} \ \textless \ 0 }$
Introduction
In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. They are used to describe the relationship between different quantities and are essential in various mathematical operations, including algebra, geometry, and calculus. In this article, we will focus on solving the inequality 9 - x/4 < 0, which is a simple yet important example of an inequality.
Understanding the Inequality
The given inequality is 9 - x/4 < 0. To solve this inequality, we need to isolate the variable x. The first step is to get rid of the fraction by multiplying both sides of the inequality by 4. This will eliminate the fraction and make it easier to solve.
Multiplying Both Sides by 4
Multiplying both sides of the inequality by 4 gives us:
4(9 - x/4) < 4(0)
Using the distributive property, we can expand the left-hand side of the inequality:
36 - x < 0
Isolating the Variable x
Now, we need to isolate the variable x. To do this, we can add x to both sides of the inequality, which will give us:
36 < x
However, this is not the correct solution. We need to subtract 36 from both sides of the inequality to get:
0 < x - 36
Adding 36 to Both Sides
Adding 36 to both sides of the inequality gives us:
36 < x - 36 + 36
Simplifying the right-hand side of the inequality, we get:
36 < x
However, this is still not the correct solution. We need to add 36 to both sides of the inequality to get:
36 + 36 < x + 36
Simplifying the left-hand side of the inequality, we get:
72 < x
Conclusion
In conclusion, the solution to the inequality 9 - x/4 < 0 is x > 72. This means that any value of x greater than 72 will satisfy the inequality.
Step-by-Step Solution
Here is the step-by-step solution to the inequality:
- Multiply both sides of the inequality by 4 to eliminate the fraction: 4(9 - x/4) < 4(0) 36 - x < 0
- Add x to both sides of the inequality to isolate the variable x: 36 < x
- Add 36 to both sides of the inequality to get the correct solution: 36 + 36 < x + 36 72 < x
Final Answer
The final answer to the inequality 9 - x/4 < 0 is x > 72.
Frequently Asked Questions
Q: What is the solution to the inequality 9 - x/4 < 0?
A: The solution to the inequality 9 - x/4 < 0 is x > 72.
Q: How do I solve the inequality 9 - x/4 < 0?
A: To solve the inequality 9 - x/4 < 0, you need to multiply both sides of the inequality by 4, add x to both sides, and then add 36 to both sides.
Q: What is the difference between an inequality and an equation?
A: An inequality is a statement that compares two or more expressions using a mathematical symbol such as <, >, ≤, or ≥. An equation is a statement that states that two or more expressions are equal.
Conclusion
In conclusion, solving the inequality 9 - x/4 < 0 requires careful manipulation of the inequality to isolate the variable x. By following the step-by-step solution outlined in this article, you can easily solve the inequality and find the correct solution.
Additional Resources
For more information on solving inequalities, you can refer to the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Wolfram Alpha: Solving Inequalities
Final Thoughts
Solving inequalities is an essential skill in mathematics that requires practice and patience. By following the step-by-step solution outlined in this article, you can easily solve the inequality 9 - x/4 < 0 and find the correct solution. Remember to always check your work and verify your solution to ensure that it is correct.
Introduction
Solving inequalities can be a challenging task, especially for those who are new to mathematics. However, with practice and patience, anyone can master the art of solving inequalities. In this article, we will provide a comprehensive Q&A guide to help you understand and solve inequalities.
Q: What is an inequality?
A: An inequality is a statement that compares two or more expressions using a mathematical symbol such as <, >, ≤, or ≥.
Q: What are the different types of inequalities?
A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities are inequalities that can be written in the form ax + b < c, where a, b, and c are constants. Quadratic inequalities are inequalities that can be written in the form ax^2 + bx + c < 0, where a, b, and c are constants.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, you need to isolate the variable x. This can be done by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: How do I solve a quadratic inequality?
A: To solve a quadratic inequality, you need to find the roots of the quadratic equation and then use the roots to determine the intervals where the inequality is true.
Q: What is the difference between an inequality and an equation?
A: An inequality is a statement that compares two or more expressions using a mathematical symbol such as <, >, ≤, or ≥. An equation is a statement that states that two or more expressions are equal.
Q: How do I determine the solution to an inequality?
A: To determine the solution to an inequality, you need to find the values of x that make the inequality true. This can be done by using the methods outlined above, such as adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the solution to the inequality 9 - x/4 < 0?
A: The solution to the inequality 9 - x/4 < 0 is x > 72.
Q: How do I graph an inequality?
A: To graph an inequality, you need to draw a number line and mark the values of x that make the inequality true. You can use a solid line to represent the inequality, or a dashed line to represent the inequality.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + b < 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 < 0, where a, b, and c are constants.
Q: How do I solve a system of inequalities?
A: To solve a system of inequalities, you need to find the values of x that make all of the inequalities true. This can be done by using the methods outlined above, such as adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the solution to the system of inequalities x + y < 2 and x - y > 1?
A: The solution to the system of inequalities x + y < 2 and x - y > 1 is x > 1 and y < 1.
Q: How do I determine the solution to a system of inequalities?
A: To determine the solution to a system of inequalities, you need to find the values of x and y that make all of the inequalities true. This can be done by using the methods outlined above, such as adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the difference between a linear inequality and a quadratic inequality in three variables?
A: A linear inequality in three variables is an inequality that can be written in the form ax + by + cz < d, where a, b, c, and d are constants. A quadratic inequality in three variables is an inequality that can be written in the form ax^2 + by^2 + cz^2 + 2axy + 2byz + 2czx < 0, where a, b, c, and d are constants.
Q: How do I solve a linear inequality in three variables?
A: To solve a linear inequality in three variables, you need to isolate the variables x, y, and z. This can be done by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: How do I solve a quadratic inequality in three variables?
A: To solve a quadratic inequality in three variables, you need to find the roots of the quadratic equation and then use the roots to determine the intervals where the inequality is true.
Q: What is the solution to the inequality x + y + z < 3?
A: The solution to the inequality x + y + z < 3 is x < 3 - y - z.
Q: How do I determine the solution to a quadratic inequality in three variables?
A: To determine the solution to a quadratic inequality in three variables, you need to find the values of x, y, and z that make the inequality true. This can be done by using the methods outlined above, such as adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Conclusion
In conclusion, solving inequalities is an essential skill in mathematics that requires practice and patience. By following the step-by-step solutions outlined in this article, you can easily solve inequalities and find the correct solution. Remember to always check your work and verify your solution to ensure that it is correct.
Additional Resources
For more information on solving inequalities, you can refer to the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Wolfram Alpha: Solving Inequalities
Final Thoughts
Solving inequalities is an essential skill in mathematics that requires practice and patience. By following the step-by-step solutions outlined in this article, you can easily solve inequalities and find the correct solution. Remember to always check your work and verify your solution to ensure that it is correct.