Solve The Inequality: $\[ -5m - 6 \geq 24 \\]

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Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. We will use the inequality −5m−6≥24-5m - 6 \geq 24 as an example to demonstrate the steps involved in solving linear inequalities.

What are Linear Inequalities?

Linear inequalities are inequalities that can be written in the form of a linear equation. They are of 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. Linear inequalities can be solved using various methods, including algebraic manipulation, graphical representation, and numerical methods.

Steps to Solve Linear Inequalities

To solve a linear inequality, we need to follow these steps:

Step 1: Write the Inequality in the Standard Form

The first step in solving a linear inequality is to write it in the standard form, which is ax+b≥cax + b \geq c or ax+b≤cax + b \leq c. In our example, the inequality is already in the standard form: −5m−6≥24-5m - 6 \geq 24.

Step 2: Add or Subtract the Same Value to Both Sides

The next step is to add or subtract the same value to both sides of the inequality to isolate the variable. In our example, we can add 6 to both sides of the inequality to get: −5m≥30-5m \geq 30.

Step 3: Divide Both Sides by the Coefficient of the Variable

Once we have isolated the variable, we can divide both sides of the inequality by the coefficient of the variable to solve for the variable. In our example, we can divide both sides of the inequality by -5 to get: m≤−6m \leq -6.

Step 4: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. In our example, the solution is m≤−6m \leq -6, which can be written in interval notation as (−∞,−6](-\infty, -6].

Example: Solving the Inequality −5m−6≥24-5m - 6 \geq 24

Now that we have gone through the steps involved in solving linear inequalities, let's apply these steps to solve the inequality −5m−6≥24-5m - 6 \geq 24.

Step 1: Write the Inequality in the Standard Form

The inequality is already in the standard form: −5m−6≥24-5m - 6 \geq 24.

Step 2: Add or Subtract the Same Value to Both Sides

We can add 6 to both sides of the inequality to get: −5m≥30-5m \geq 30.

Step 3: Divide Both Sides by the Coefficient of the Variable

We can divide both sides of the inequality by -5 to get: m≤−6m \leq -6.

Step 4: Write the Solution in Interval Notation

The solution is m≤−6m \leq -6, which can be written in interval notation as (−∞,−6](-\infty, -6].

Conclusion

Solving linear inequalities is an essential skill in mathematics that can be applied to various problems in algebra, geometry, and other branches of mathematics. By following the steps outlined in this article, we can solve linear inequalities with ease. Remember to write the inequality in the standard form, add or subtract the same value to both sides, divide both sides by the coefficient of the variable, and write the solution in interval notation.

Frequently Asked Questions

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow the steps outlined in this article: write the inequality in the standard form, add or subtract the same value to both sides, divide both sides by the coefficient of the variable, and write the solution in interval notation.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality in a compact form. It consists of a pair of numbers, separated by a comma, that represent the lower and upper bounds of the solution.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Mathematics for the Nonmathematician" by Morris Kline
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Further Reading

  • [1] "Solving Linear Inequalities" by Math Open Reference
  • [2] "Linear Inequalities" by Khan Academy
  • [3] "Solving Linear Inequalities" by Purplemath

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Introduction

Solving linear inequalities can be a challenging task, especially for those who are new to algebra. However, with the right guidance and practice, anyone can become proficient in solving linear inequalities. In this article, we will answer some of the most frequently asked questions about solving linear inequalities.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation. It is an expression that compares two values, indicating that one is greater than, less than, or equal to the other.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow these steps:

  1. Write the inequality in the standard form.
  2. Add or subtract the same value to both sides.
  3. Divide both sides by the coefficient of the variable.
  4. Write the solution in interval notation.

Q: What is the standard form of a linear inequality?

A: The standard form of a linear inequality is 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 add or subtract the same value to both sides of a linear inequality?

A: To add or subtract the same value to both sides of a linear inequality, you need to perform the same operation on both sides. For example, if you have the inequality x+3≥5x + 3 \geq 5, you can subtract 3 from both sides to get x≥2x \geq 2.

Q: How do I divide both sides of a linear inequality by the coefficient of the variable?

A: To divide both sides of a linear inequality by the coefficient of the variable, you need to perform the division operation on both sides. For example, if you have the inequality 2x≥62x \geq 6, you can divide both sides by 2 to get x≥3x \geq 3.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality in a compact form. It consists of a pair of numbers, separated by a comma, that represent the lower and upper bounds of the solution.

Q: How do I write the solution to a linear inequality in interval notation?

A: To write the solution to a linear inequality in interval notation, you need to identify the lower and upper bounds of the solution. For example, if you have the inequality x≥2x \geq 2, the solution is [2,∞)[2, \infty).

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

A: Yes, you can use a calculator to solve linear inequalities. However, it is always a good idea to check your work by hand to ensure that the solution is correct.

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

A: Yes, you can use a graphing calculator to solve linear inequalities. Graphing calculators can help you visualize the solution to an inequality and can also be used to check your work.

Conclusion

Solving linear inequalities can be a challenging task, but with the right guidance and practice, anyone can become proficient. By following the steps outlined in this article and practicing with different types of linear inequalities, you can become a master of solving linear inequalities.

Frequently Asked Questions (FAQs)

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

A: A linear inequality is an inequality that can be written in the form of a linear equation, while a quadratic inequality is an inequality that can be written in the form of a quadratic equation.

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

A: No, you cannot use the same steps to solve quadratic inequalities as you do to solve linear inequalities. Quadratic inequalities require a different set of steps to solve.

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

A: Yes, you can use a calculator to solve quadratic inequalities. However, it is always a good idea to check your work by hand to ensure that the solution is correct.

Q: Can I use a graphing calculator to solve quadratic inequalities?

A: Yes, you can use a graphing calculator to solve quadratic inequalities. Graphing calculators can help you visualize the solution to an inequality and can also be used to check your work.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Mathematics for the Nonmathematician" by Morris Kline
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Further Reading

  • [1] "Solving Linear Inequalities" by Math Open Reference
  • [2] "Linear Inequalities" by Khan Academy
  • [3] "Solving Linear Inequalities" by Purplemath