Solve For $h$.$3 \geq -6(h+2) - 9$Write Your Answer Using A Decimal, Whole Number, Or Fraction In Lowest Terms: □ \square □

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Introduction

Linear inequalities are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities, specifically the inequality 3≥−6(h+2)−93 \geq -6(h+2) - 9. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding the Inequality

Before we dive into solving the inequality, let's take a closer look at the given expression: 3≥−6(h+2)−93 \geq -6(h+2) - 9. This is a linear inequality, which means it is an inequality that can be written in the form ax+b≥cax + b \geq c, where aa, bb, and cc are constants.

Key Components of the Inequality

  • The variable hh is the subject of the inequality.
  • The coefficient of hh is −6-6, which means that for every unit increase in hh, the value of the expression decreases by 66.
  • The constant term is −9-9, which is added to the expression.

Solving the Inequality

To solve the inequality, we need to isolate the variable hh on one side of the inequality sign. We can do this by performing a series of algebraic operations.

Step 1: Distribute the Negative 6

The first step is to distribute the negative 6 to the terms inside the parentheses:

3≥−6(h+2)−93 \geq -6(h+2) - 9

3≥−6h−12−93 \geq -6h - 12 - 9

3≥−6h−213 \geq -6h - 21

Step 2: Add 21 to Both Sides

Next, we add 21 to both sides of the inequality to get rid of the negative term:

3+21≥−6h−21+213 + 21 \geq -6h - 21 + 21

24≥−6h24 \geq -6h

Step 3: Divide Both Sides by -6

Now, we divide both sides of the inequality by −6-6 to isolate the variable hh. When we divide by a negative number, we need to reverse the direction of the inequality sign:

24−6≤−6h−6\frac{24}{-6} \leq \frac{-6h}{-6}

−4≤h-4 \leq h

Conclusion

In this article, we solved the linear inequality 3≥−6(h+2)−93 \geq -6(h+2) - 9 step by step. We distributed the negative 6, added 21 to both sides, and finally divided both sides by −6-6 to isolate the variable hh. The solution to the inequality is h≤−4h \leq -4.

Frequently Asked Questions

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

A: A linear equation is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. A linear inequality, on the other hand, is an inequality that can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c.

Q: How do I know which direction to flip the inequality sign when dividing by a negative number?

A: When dividing by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality a≥ba \geq b and you divide both sides by −c-c, the resulting inequality will be a≤ba \leq b.

Q: Can I use the same steps to solve a linear inequality with a variable on both sides?

A: No, you cannot use the same steps to solve a linear inequality with a variable on both sides. In such cases, you need to use a different approach, such as isolating the variable on one side of the inequality sign.

Final Thoughts

Solving linear inequalities requires a clear understanding of the concept and a step-by-step approach. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to always check your work and verify the solution to ensure that it is correct.

Additional Resources

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

References

  • [1] "Linear Inequalities" by Khan Academy
  • [2] "Linear Inequality Solver" by Mathway
  • [3] "Linear Inequality Solver" by Wolfram Alpha

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Introduction

Linear inequalities are a fundamental concept in algebra, and solving them can be a challenging task for many students. In this article, we will address some of the most frequently asked questions about linear inequalities, providing clear and concise answers to help you better understand this concept.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that 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.

Q: How do I know which direction to flip the inequality sign when dividing by a negative number?

A: When dividing by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality a≥ba \geq b and you divide both sides by −c-c, the resulting inequality will be a≤ba \leq b.

Q: Can I use the same steps to solve a linear inequality with a variable on both sides?

A: No, you cannot use the same steps to solve a linear inequality with a variable on both sides. In such cases, you need to use a different approach, such as isolating the variable on one side of the inequality sign.

Q: How do I solve a linear inequality with a fraction?

A: To solve a linear inequality with a fraction, you need to follow the same steps as solving a linear inequality with a variable. First, multiply both sides of the inequality by the denominator to eliminate the fraction. Then, solve the resulting inequality.

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

A: Yes, you can use a calculator to solve a linear inequality. However, it's essential to understand the concept and the steps involved in solving a linear inequality, as using a calculator alone may not provide a complete understanding of the solution.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to follow these steps:

  1. Identify the inequality sign.
  2. Determine the direction of the inequality.
  3. Plot a point on the number line that satisfies the inequality.
  4. Draw an arrow on the number line to indicate the direction of the inequality.

Q: Can I use a linear inequality to solve a system of linear equations?

A: Yes, you can use a linear inequality to solve a system of linear equations. However, it's essential to understand the concept and the steps involved in solving a system of linear equations, as using a linear inequality alone may not provide a complete solution.

Conclusion

In this article, we addressed some of the most frequently asked questions about linear inequalities, providing clear and concise answers to help you better understand this concept. Whether you're a student or a teacher, we hope this article has been helpful in clarifying your understanding of linear inequalities.

Additional Resources

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

References

  • [1] "Linear Inequalities" by Khan Academy
  • [2] "Linear Inequality Solver" by Mathway
  • [3] "Linear Inequality Solver" by Wolfram Alpha

Final Thoughts

Linear inequalities are a fundamental concept in algebra, and solving them requires a clear understanding of the concept and the steps involved. By following the steps outlined in this article and using the additional resources provided, you can become proficient in solving linear inequalities and apply this knowledge to real-world problems.

Common Mistakes to Avoid

  • Not following the correct order of operations when solving a linear inequality.
  • Not reversing the direction of the inequality sign when dividing by a negative number.
  • Not isolating the variable on one side of the inequality sign when solving a linear inequality with a variable on both sides.
  • Not using a calculator to check the solution to a linear inequality.

Tips and Tricks

  • Use a number line to visualize the solution to a linear inequality.
  • Check your work by plugging in values that satisfy the inequality.
  • Use a calculator to check the solution to a linear inequality.
  • Practice, practice, practice! The more you practice solving linear inequalities, the more comfortable you will become with the concept.