Which Represents The Solution Set Of The Inequality 5 X − 9 ≤ 21 5x - 9 \leq 21 5 X − 9 ≤ 21 ?A. X ≤ 12 5 X \leq \frac{12}{5} X ≤ 5 12 ​ B. X ≥ 12 5 X \geq \frac{12}{5} X ≥ 5 12 ​ C. X ≥ 6 X \geq 6 X ≥ 6 D. X ≤ 6 X \leq 6 X ≤ 6

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Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities of the form ax≤bax \leq b, where aa and bb are constants. We will use the given inequality 5x−9≤215x - 9 \leq 21 as a case study to demonstrate the step-by-step process of solving linear inequalities.

Understanding the Inequality

The given inequality is 5x−9≤215x - 9 \leq 21. To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. The first step is to add 99 to both sides of the inequality to get rid of the negative term.

Adding 9 to Both Sides

5x - 9 + 9 ≤ 21 + 9
5x ≤ 30

Isolating the Variable

The next step is to isolate the variable xx by dividing both sides of the inequality by the coefficient of xx, which is 55.

Dividing Both Sides by 5

(5x) / 5 ≤ 30 / 5
x ≤ 6

Analyzing the Solution Set

The solution set of the inequality 5x−9≤215x - 9 \leq 21 is represented by the inequality x≤6x \leq 6. This means that any value of xx that is less than or equal to 66 satisfies the inequality.

Comparing with the Options

Now, let's compare the solution set x≤6x \leq 6 with the given options:

  • A. x≤125x \leq \frac{12}{5}: This option is incorrect because the solution set is x≤6x \leq 6, not x≤125x \leq \frac{12}{5}.
  • B. x≥125x \geq \frac{12}{5}: This option is incorrect because the solution set is x≤6x \leq 6, not x≥125x \geq \frac{12}{5}.
  • C. x≥6x \geq 6: This option is incorrect because the solution set is x≤6x \leq 6, not x≥6x \geq 6.
  • D. x≤6x \leq 6: This option is correct because the solution set is indeed x≤6x \leq 6.

Conclusion

In conclusion, the solution set of the inequality 5x−9≤215x - 9 \leq 21 is represented by the inequality x≤6x \leq 6. This means that any value of xx that is less than or equal to 66 satisfies the inequality. By following the step-by-step process of solving linear inequalities, we can easily determine the solution set and compare it with the given options.

Frequently Asked Questions

Q: What is the solution set of the inequality 5x−9≤215x - 9 \leq 21?

A: The solution set is x≤6x \leq 6.

Q: How do I solve linear inequalities?

A: To solve linear inequalities, you need to isolate the variable on one side of the inequality sign by adding or subtracting the same value from both sides, and then dividing both sides by the coefficient of the variable.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear inequality is an inequality that involves a linear expression, whereas a linear equation is an equation that involves a linear expression. In a linear inequality, the inequality sign is used to indicate the relationship between the variables, whereas in a linear equation, the equality sign is used to indicate the relationship between the variables.

Q: Can I use the same steps to solve quadratic inequalities?

A: No, the steps to solve quadratic inequalities are different from the steps to solve linear inequalities. Quadratic inequalities involve quadratic expressions, and the steps to solve them involve factoring, completing the square, or using the quadratic formula.

Final Thoughts

Solving linear inequalities is a crucial skill for students to master, and it has numerous applications in real-life situations. By following the step-by-step process of solving linear inequalities, we can easily determine the solution set and compare it with the given options. In this article, we used the inequality 5x−9≤215x - 9 \leq 21 as a case study to demonstrate the step-by-step process of solving linear inequalities. We hope that this article has provided valuable insights and knowledge on solving linear inequalities.

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Introduction

In our previous article, we discussed the step-by-step process of solving linear inequalities. However, we understand that sometimes, it's not enough to just provide a general guide. Students often have specific questions and concerns that need to be addressed. In this article, we will provide a Q&A guide to help students better understand and solve linear inequalities.

Q&A Guide

Q: What is the difference between a linear inequality and a linear equation?

A: A linear inequality is an inequality that involves a linear expression, whereas a linear equation is an equation that involves a linear expression. In a linear inequality, the inequality sign is used to indicate the relationship between the variables, whereas in a linear equation, the equality sign is used to indicate the relationship between the variables.

Q: How do I solve linear inequalities?

A: To solve linear inequalities, you need to isolate the variable on one side of the inequality sign by adding or subtracting the same value from both sides, and then dividing both sides by the coefficient of the variable.

Q: What is the order of operations when solving linear inequalities?

A: The order of operations when solving linear inequalities is the same as when solving linear equations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use the same steps to solve quadratic inequalities?

A: No, the steps to solve quadratic inequalities are different from the steps to solve linear inequalities. Quadratic inequalities involve quadratic expressions, and the steps to solve them involve factoring, completing the square, or using the quadratic formula.

Q: How do I determine the solution set of a linear inequality?

A: To determine the solution set of a linear inequality, you need to isolate the variable on one side of the inequality sign and then determine the values of the variable that satisfy the inequality.

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 sign, such as <,>,≠<, >, \neq. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as ≤,≥,=\leq, \geq, =. Strict inequalities have no equal sign, while non-strict inequalities have an equal sign.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Examples

Example 1: Solving a Linear Inequality

Solve the linear inequality 2x+5≤112x + 5 \leq 11.

2x + 5 - 5 ≤ 11 - 5
2x ≤ 6
x ≤ 3

Example 2: Solving a Quadratic Inequality

Solve the quadratic inequality x2+4x+4≥0x^2 + 4x + 4 \geq 0.

(x + 2)^2 ≥ 0
x + 2 ≥ 0 or x + 2 ≤ 0
x ≥ -2 or x ≤ -2

Conclusion

In conclusion, solving linear inequalities is a crucial skill for students to master. By following the step-by-step process of solving linear inequalities and using the Q&A guide provided in this article, students can better understand and solve linear inequalities. Remember to always isolate the variable on one side of the inequality sign and determine the values of the variable that satisfy the inequality.

Frequently Asked Questions

Q: What is the solution set of the inequality 2x+5≤112x + 5 \leq 11?

A: The solution set is x≤3x \leq 3.

Q: How do I determine the solution set of a linear inequality?

A: To determine the solution set of a linear inequality, you need to isolate the variable on one side of the inequality sign and then determine the values of the variable that satisfy the inequality.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

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 sign, such as <,>,≠<, >, \neq. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as ≤,≥,=\leq, \geq, =. Strict inequalities have no equal sign, while non-strict inequalities have an equal sign.

Final Thoughts

Solving linear inequalities is a crucial skill for students to master. By following the step-by-step process of solving linear inequalities and using the Q&A guide provided in this article, students can better understand and solve linear inequalities. Remember to always isolate the variable on one side of the inequality sign and determine the values of the variable that satisfy the inequality.