Which Sentence Is Represented By 67 Z + 8.9 ≤ − 302 67z + 8.9 \leq -302 67 Z + 8.9 ≤ − 302 ?A. Sixty-seven Times A Number Is Less Than Negative Three Hundred Two.B. Sixty-seven Times A Number, Plus Eight And Nine-tenths, Is Not Greater Than Negative Three Hundred Two.C.
In algebra, inequalities are used to represent relationships between variables and constants. They are essential in solving problems and making decisions in various fields, including mathematics, science, and engineering. In this article, we will explore the concept of inequalities and how to represent them in algebraic form.
What are Inequalities?
Inequalities are mathematical statements that compare two expressions, indicating whether one is greater than, less than, or equal to the other. They are used to represent relationships between variables and constants, and are essential in solving problems and making decisions in various fields.
Types of Inequalities
There are several types of inequalities, including:
- Linear inequalities: These are inequalities that involve linear expressions, such as 2x + 3 ≤ 5.
- Quadratic inequalities: These are inequalities that involve quadratic expressions, such as x^2 + 4x + 4 ≥ 0.
- Absolute value inequalities: These are inequalities that involve absolute value expressions, such as |x| ≤ 3.
Representing Inequalities in Algebra
In algebra, inequalities are represented using symbols such as ≤, ≥, <, and >. The inequality symbol is used to indicate the relationship between the two expressions.
- Less than or equal to: ≤
- Greater than or equal to: ≥
- Less than: <
- Greater than: >
For example, the inequality 2x + 3 ≤ 5 can be represented as:
2x + 3 ≤ 5
This inequality states that 2x + 3 is less than or equal to 5.
Solving Inequalities
Solving inequalities involves finding the values of the variable that satisfy the inequality. There are several methods for solving inequalities, including:
- Adding or subtracting a constant: This involves adding or subtracting a constant to both sides of the inequality to isolate the variable.
- Multiplying or dividing by a constant: This involves multiplying or dividing both sides of the inequality by a constant to isolate the variable.
- Using inverse operations: This involves using inverse operations, such as addition and subtraction, to isolate the variable.
For example, to solve the inequality 2x + 3 ≤ 5, we can subtract 3 from both sides to get:
2x ≤ 2
Then, we can divide both sides by 2 to get:
x ≤ 1
Which Sentence is Represented by ?
Now that we have a basic understanding of inequalities, let's look at the given inequality:
This inequality states that 67z + 8.9 is less than or equal to -302.
A. Sixty-seven times a number is less than negative three hundred two.
This sentence is not represented by the given inequality. The inequality states that 67z + 8.9 is less than or equal to -302, not that 67z is less than -302.
B. Sixty-seven times a number, plus eight and nine-tenths, is not greater than negative three hundred two.
This sentence is represented by the given inequality. The inequality states that 67z + 8.9 is less than or equal to -302, which means that 67z + 8.9 is not greater than -302.
C. None of the above.
This option is incorrect. The correct option is B.
Conclusion
In conclusion, the given inequality is represented by the sentence "Sixty-seven times a number, plus eight and nine-tenths, is not greater than negative three hundred two." This sentence accurately represents the relationship between the two expressions in the inequality.
Frequently Asked Questions
Q: What is an inequality in algebra?
A: An inequality in algebra is a mathematical statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other.
Q: What are the different types of inequalities?
A: There are several types of inequalities, including linear inequalities, quadratic inequalities, and absolute value inequalities.
Q: How do I solve an inequality?
A: To solve an inequality, you can use methods such as adding or subtracting a constant, multiplying or dividing by a constant, and using inverse operations.
Q: What is the difference between an inequality and an equation?
A: An inequality is a mathematical statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other. An equation is a mathematical statement that states that two expressions are equal.
Q: How do I determine which sentence is represented by a given inequality?
In this article, we will answer some of the most frequently asked questions about inequalities in algebra.
Q: What is an inequality in algebra?
A: An inequality in algebra is a mathematical statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other.
Q: What are the different types of inequalities?
A: There are several types of inequalities, including:
- Linear inequalities: These are inequalities that involve linear expressions, such as 2x + 3 ≤ 5.
- Quadratic inequalities: These are inequalities that involve quadratic expressions, such as x^2 + 4x + 4 ≥ 0.
- Absolute value inequalities: These are inequalities that involve absolute value expressions, such as |x| ≤ 3.
Q: How do I solve an inequality?
A: To solve an inequality, you can use methods such as:
- Adding or subtracting a constant: This involves adding or subtracting a constant to both sides of the inequality to isolate the variable.
- Multiplying or dividing by a constant: This involves multiplying or dividing both sides of the inequality by a constant to isolate the variable.
- Using inverse operations: This involves using inverse operations, such as addition and subtraction, to isolate the variable.
Q: What is the difference between an inequality and an equation?
A: An inequality is a mathematical statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other. An equation is a mathematical statement that states that two expressions are equal.
Q: How do I determine which sentence is represented by a given inequality?
A: To determine which sentence is represented by a given inequality, you need to carefully read the inequality and identify the relationship between the two expressions. Then, you can compare the inequality to the given sentences and choose the one that accurately represents the relationship.
Q: Can I use the same methods to solve inequalities as I do to solve equations?
A: Yes, you can use many of the same methods to solve inequalities as you do to solve equations. However, you need to be careful when multiplying or dividing both sides of an inequality by a negative number, as this can change the direction of the inequality.
Q: How do I know which direction to use when solving an inequality?
A: When solving an inequality, you need to use the same direction as the original inequality. For example, if the original inequality is 2x + 3 ≤ 5, you need to use the ≤ symbol when solving the inequality.
Q: Can I use inequalities to solve problems in real-life situations?
A: Yes, inequalities can be used to solve problems in real-life situations. For example, you can use inequalities to determine the maximum or minimum value of a quantity, or to compare the values of two or more quantities.
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to use a closed circle to represent the endpoint of the inequality, and an open circle to represent the endpoint of the inequality that is not included. For example, the inequality x ≥ 2 would be graphed as a closed circle at x = 2, and an open circle at x = 1.
Q: Can I use inequalities to solve problems in calculus?
A: Yes, inequalities can be used to solve problems in calculus. For example, you can use inequalities to determine the maximum or minimum value of a function, or to compare the values of two or more functions.
Conclusion
In conclusion, inequalities are an important concept in algebra that can be used to solve problems in a variety of situations. By understanding how to solve inequalities and how to use them to solve problems, you can become a more confident and proficient math student.
Additional Resources
If you are looking for additional resources to help you learn about inequalities, here are a few suggestions:
- Textbooks: There are many textbooks available that cover the topic of inequalities in algebra.
- Online resources: There are many online resources available that provide tutorials and examples on how to solve inequalities.
- Practice problems: Practice problems are a great way to reinforce your understanding of inequalities and to build your problem-solving skills.
Final Thoughts
Inequalities are an important concept in algebra that can be used to solve problems in a variety of situations. By understanding how to solve inequalities and how to use them to solve problems, you can become a more confident and proficient math student. Remember to practice regularly and to seek help when you need it. With practice and patience, you can master the concept of inequalities and become a skilled math student.