Solve The Inequalities:1. { -9n \ \textless \ 54$}$ 2. { N - 8 \leq -15$}$
In mathematics, inequalities are a fundamental concept that helps us compare two or more values. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving two types of inequalities: linear inequalities and compound inequalities.
What are Inequalities?
Inequalities are mathematical statements that compare two or more values using the symbols <, >, ≤, or ≥. For example, the statement "x > 5" means that x is greater than 5, while the statement "x ≤ 5" means that x is less than or equal to 5.
Types of Inequalities
There are two main types of inequalities: linear inequalities and compound inequalities.
- Linear Inequalities: A linear inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants. For example, the inequality 2x + 3 < 7 is a linear inequality.
- Compound Inequalities: A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". For example, the inequality 2x + 3 < 7 and x - 2 > 3 is a compound inequality.
Solving Linear Inequalities
To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. We can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Solving the Inequality
To solve the inequality , we need to isolate the variable n. We can do this by dividing both sides of the inequality by -9.
-9n < 54
n > -54/9
n > -6
Therefore, the solution to the inequality is n > -6.
Solving the Inequality {n - 8 \leq -15$}$
To solve the inequality {n - 8 \leq -15$}$, we need to isolate the variable n. We can do this by adding 8 to both sides of the inequality.
n - 8 + 8 ≤ -15 + 8
n ≤ -7
Therefore, the solution to the inequality {n - 8 \leq -15$}$ is n ≤ -7.
Conclusion
Solving inequalities is an essential skill in mathematics that helps us compare two or more values. In this article, we have focused on solving two types of inequalities: linear inequalities and compound inequalities. We have also provided step-by-step solutions to two inequalities: and {n - 8 \leq -15$}$. By following the steps outlined in this article, you can solve any linear inequality and compound inequality.
Frequently Asked Questions
Q: What is the difference between a linear inequality and a compound 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 compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or".
Q: How do I solve a linear inequality?
A: To solve a linear 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, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the solution to the inequality ?
A: The solution to the inequality is n > -6.
Q: What is the solution to the inequality {n - 8 \leq -15$}$?
A: The solution to the inequality {n - 8 \leq -15$}$ is n ≤ -7.
References
Additional Resources
- Inequalities
- Linear Inequalities
- Compound Inequalities
Solving Inequalities: A Q&A Guide =====================================
In our previous article, we discussed the basics of solving inequalities, including linear inequalities and compound inequalities. In this article, we will provide a Q&A guide to help you better understand and solve inequalities.
Q: What is the difference between a linear inequality and a compound 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 compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or".
Q: How do I solve a linear inequality?
A: To solve a linear 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, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the solution to the inequality ?
A: The solution to the inequality is n > -6.
Q: What is the solution to the inequality {n - 8 \leq -15$}$?
A: The solution to the inequality {n - 8 \leq -15$}$ is n ≤ -7.
Q: How do I solve a compound inequality?
A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. For example, if you have the compound inequality 2x + 3 < 7 and x - 2 > 3, you would first solve each inequality separately and then combine the solutions.
Q: What is the solution to the compound inequality 2x + 3 < 7 and x - 2 > 3?
A: To solve the compound inequality 2x + 3 < 7 and x - 2 > 3, you would first solve each inequality separately.
2x + 3 < 7
2x < 4
x < 2
x - 2 > 3
x > 5
Since the two inequalities are joined by the word "and", you would need to find the intersection of the two solutions. In this case, the solution to the compound inequality is x > 5.
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you would first identify the solution to the inequality. Then, you would plot a point on the number line that represents the solution. If the inequality is of the form x > a, you would plot a point to the right of a. If the inequality is of the form x < a, you would plot a point to the left of a.
Q: How do I graph the inequality x > 5 on a number line?
A: To graph the inequality x > 5 on a number line, you would first identify the solution to the inequality, which is x > 5. Then, you would plot a point on the number line that represents the solution. Since the inequality is of the form x > a, you would plot a point to the right of 5.
Q: What is the difference between a strict inequality and a non-strict inequality?
A: A strict inequality is an inequality that is written with a strict inequality symbol, such as < or >. A non-strict inequality is an inequality that is written with a non-strict inequality symbol, such as ≤ or ≥.
Q: How do I solve a non-strict inequality?
A: To solve a non-strict inequality, you would follow the same steps as solving a strict inequality. However, you would also include the value that makes the inequality true.
Q: What is the solution to the non-strict inequality x ≤ 5?
A: The solution to the non-strict inequality x ≤ 5 is x ≤ 5.
Conclusion
Solving inequalities is an essential skill in mathematics that helps us compare two or more values. In this article, we have provided a Q&A guide to help you better understand and solve inequalities. We have also provided examples and step-by-step solutions to help you practice solving inequalities.
Frequently Asked Questions
Q: What is the difference between a linear inequality and a compound 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 compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or".
Q: How do I solve a linear inequality?
A: To solve a linear 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, or by multiplying or dividing both sides of the inequality by the same non-zero value.
Q: What is the solution to the inequality ?
A: The solution to the inequality is n > -6.
Q: What is the solution to the inequality {n - 8 \leq -15$}$?
A: The solution to the inequality {n - 8 \leq -15$}$ is n ≤ -7.