Which Statements Are True About The Solution Of 15 ≥ 22 + X 15 \geq 22 + X 15 ≥ 22 + X ? Select Three Options.A. X ≥ − 7 X \geq -7 X ≥ − 7 B. X ≤ − 7 X \leq -7 X ≤ − 7 C. The Graph Has A Closed Circle. D. -6 Is Part Of The Solution. E. -7 Is Part Of The Solution.

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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 the linear inequality 15≥22+x15 \geq 22 + x. We will explore the different options provided and determine which ones are true about the solution.

Understanding Linear Inequalities

A linear inequality is an inequality that involves a linear expression. It can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c, where aa, bb, and cc are constants, and xx is the variable. In this case, we have the inequality 15≥22+x15 \geq 22 + x.

Solving the Inequality

To solve the inequality, we need to isolate the variable xx. We can start by subtracting 22 from both sides of the inequality:

15−22≥x15 - 22 \geq x

This simplifies to:

−7≥x-7 \geq x

Analyzing the Options

Now that we have solved the inequality, let's analyze the options provided:

Option A: x≥−7x \geq -7

This option is incorrect because the inequality we solved is −7≥x-7 \geq x, not x≥−7x \geq -7. The correct inequality indicates that xx is less than or equal to -7.

Option B: x≤−7x \leq -7

This option is correct because the inequality we solved is −7≥x-7 \geq x, which is equivalent to x≤−7x \leq -7.

Option C: The graph has a closed circle.

This option is incorrect because the inequality we solved is a greater-than-or-equal-to inequality, which is represented by a closed circle on the number line. However, the correct inequality is −7≥x-7 \geq x, not x≥−7x \geq -7.

Option D: -6 is part of the solution.

This option is incorrect because the inequality we solved is −7≥x-7 \geq x, which means that xx is less than or equal to -7. Since -6 is greater than -7, it is not part of the solution.

Option E: -7 is part of the solution.

This option is correct because the inequality we solved is −7≥x-7 \geq x, which means that xx is less than or equal to -7. Since -7 is equal to -7, it is part of the solution.

Conclusion

In conclusion, the correct options about the solution of 15≥22+x15 \geq 22 + x are:

  • Option B: x≤−7x \leq -7
  • Option E: -7 is part of the solution.

The other options are incorrect because they do not accurately represent the solution to the inequality.

Frequently Asked Questions

Q: What is the difference between a greater-than-or-equal-to inequality and a less-than-or-equal-to inequality?

A: A greater-than-or-equal-to inequality is represented by a closed circle on the number line, while a less-than-or-equal-to inequality is represented by an open circle.

Q: How do I determine if a number is part of the solution to an inequality?

A: To determine if a number is part of the solution, you need to check if it satisfies the inequality. If the number is greater than or equal to the value on the right-hand side of the inequality, then it is part of the solution.

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

A: Yes, you can use a calculator to solve linear inequalities. However, it's always a good idea to check your work by plugging in values to make sure that the solution is correct.

Final Thoughts

Solving linear inequalities is an important skill that requires practice and patience. By following the steps outlined in this article, you can master the art of solving linear inequalities and become more confident in your math skills. Remember to always check your work and to use a calculator if needed. With practice and dedication, you can become a math whiz and tackle even the toughest inequalities with ease.

Additional Resources

  • Khan Academy: Linear Inequalities
  • Mathway: Linear Inequality Solver
  • Wolfram Alpha: Linear Inequality Solver

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Mathematics for the Nonmathematician" by Morris Kline
  • "Linear Algebra and Its Applications" by Gilbert Strang

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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 provide answers to frequently asked questions about linear inequalities, including how to solve them, how to determine if a number is part of the solution, and more.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. It can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable xx. You can start by adding or subtracting the same value to both sides of the inequality, and then multiplying or dividing both sides by the same non-zero value.

Q: What is the difference between a greater-than-or-equal-to inequality and a less-than-or-equal-to inequality?

A: A greater-than-or-equal-to inequality is represented by a closed circle on the number line, while a less-than-or-equal-to inequality is represented by an open circle.

Q: How do I determine if a number is part of the solution to an inequality?

A: To determine if a number is part of the solution, you need to check if it satisfies the inequality. If the number is greater than or equal to the value on the right-hand side of the inequality, then it is part of the solution.

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

A: Yes, you can use a calculator to solve linear inequalities. However, it's always a good idea to check your work by plugging in values to make sure that the solution is correct.

Q: What is the solution to the inequality 2x+3≥52x + 3 \geq 5?

A: To solve the inequality, we need to isolate the variable xx. We can start by subtracting 3 from both sides of the inequality:

2x+3−3≥5−32x + 3 - 3 \geq 5 - 3

This simplifies to:

2x≥22x \geq 2

Next, we can divide both sides of the inequality by 2:

2x2≥22\frac{2x}{2} \geq \frac{2}{2}

This simplifies to:

x≥1x \geq 1

Therefore, the solution to the inequality is x≥1x \geq 1.

Q: What is the solution to the inequality x−2≤3x - 2 \leq 3?

A: To solve the inequality, we need to isolate the variable xx. We can start by adding 2 to both sides of the inequality:

x−2+2≤3+2x - 2 + 2 \leq 3 + 2

This simplifies to:

x≤5x \leq 5

Therefore, the solution to the inequality is x≤5x \leq 5.

Q: Can I have a negative number as a coefficient in a linear inequality?

A: Yes, you can have a negative number as a coefficient in a linear inequality. For example, the inequality −3x+2≥5-3x + 2 \geq 5 has a negative coefficient of -3.

Q: Can I have a fraction as a coefficient in a linear inequality?

A: Yes, you can have a fraction as a coefficient in a linear inequality. For example, the inequality 12x+2≥5\frac{1}{2}x + 2 \geq 5 has a fraction coefficient of 12\frac{1}{2}.

Conclusion

In conclusion, linear inequalities are an important concept in mathematics, and solving them requires practice and patience. By following the steps outlined in this article, you can master the art of solving linear inequalities and become more confident in your math skills. Remember to always check your work and to use a calculator if needed. With practice and dedication, you can become a math whiz and tackle even the toughest inequalities with ease.

Additional Resources

  • Khan Academy: Linear Inequalities
  • Mathway: Linear Inequality Solver
  • Wolfram Alpha: Linear Inequality Solver

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Mathematics for the Nonmathematician" by Morris Kline
  • "Linear Algebra and Its Applications" by Gilbert Strang