Solve The Inequality:1) $-4m \geq -4$

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Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality −4m≥−4-4m \geq -4.

Understanding the Inequality

The given inequality is −4m≥−4-4m \geq -4. This means that the product of −4-4 and mm is greater than or equal to −4-4. To solve this inequality, we need to isolate the variable mm.

Step 1: Divide Both Sides by -4

To isolate the variable mm, we need to get rid of the coefficient −4-4. We can do this by dividing both sides of the inequality by −4-4. However, when we divide or multiply both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

\frac{-4m}{-4} \geq \frac{-4}{-4}

This simplifies to:

m \leq 1

Step 2: Write the Solution in Interval Notation

The solution to the inequality m≤1m \leq 1 can be written in interval notation as (−∞,1](-\infty, 1]. This means that the value of mm can be any real number less than or equal to 11.

Step 3: Graph the Solution on a Number Line

To visualize the solution, we can graph it on a number line. The number line consists of a horizontal line with a series of equally spaced points. We can mark the point 11 on the number line and shade the region to the left of 11 to represent the solution.

Conclusion

Solving the inequality −4m≥−4-4m \geq -4 involves isolating the variable mm by dividing both sides of the inequality by −4-4. We need to reverse the direction of the inequality sign when dividing by a negative number. The solution to the inequality is m≤1m \leq 1, which can be written in interval notation as (−∞,1](-\infty, 1]. We can graph the solution on a number line to visualize the region that satisfies the inequality.

Examples and Applications

Solving inequalities is an essential skill in mathematics and has numerous applications in real-life situations. Here are a few examples:

  • Finance: In finance, inequalities are used to model investment returns and risk. For example, an investor may want to know the minimum return on investment (ROI) required to achieve a certain level of wealth.
  • Science: In science, inequalities are used to model physical phenomena such as temperature, pressure, and velocity. For example, the ideal gas law is an inequality that relates the pressure, volume, and temperature of a gas.
  • Engineering: In engineering, inequalities are used to design and optimize systems such as electrical circuits, mechanical systems, and communication networks.

Tips and Tricks

Here are a few tips and tricks to help you solve inequalities:

  • Use inverse operations: To isolate the variable, use inverse operations such as addition, subtraction, multiplication, and division.
  • Reverse the inequality sign: When dividing or multiplying both sides of an inequality by a negative number, reverse the direction of the inequality sign.
  • Check your work: Always check your work by plugging in values to ensure that the solution satisfies the inequality.

Common Mistakes

Here are a few common mistakes to avoid when solving inequalities:

  • Forgetting to reverse the inequality sign: When dividing or multiplying both sides of an inequality by a negative number, forget to reverse the direction of the inequality sign.
  • Not checking your work: Not checking your work by plugging in values can lead to incorrect solutions.
  • Not using inverse operations: Not using inverse operations such as addition, subtraction, multiplication, and division can make it difficult to isolate the variable.

Conclusion

Introduction

In our previous article, we discussed how to solve the inequality −4m≥−4-4m \geq -4. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to symbols.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by using inverse operations such as addition, subtraction, multiplication, and division.

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 ax+b≥cax + b \geq c or ax+b≤cax + b \leq c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c≥0ax^2 + bx + c \geq 0 or ax2+bx+c≤0ax^2 + bx + c \leq 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the expression to determine the solution. You can also use the quadratic formula to find the solutions to the quadratic equation.

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 x>2x > 2 or x<3x < 3. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as x≥2x \geq 2 or x≤3x \leq 3.

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

A: To graph an inequality on a number line, you need to mark the point that satisfies the inequality and then shade the region to the left or right of the point.

Q: What is the solution to the inequality x2+4x+4≥0x^2 + 4x + 4 \geq 0?

A: The solution to the inequality x2+4x+4≥0x^2 + 4x + 4 \geq 0 is x≤−2x \leq -2.

Q: What is the solution to the inequality x2+4x+5≤0x^2 + 4x + 5 \leq 0?

A: The solution to the inequality x2+4x+5≤0x^2 + 4x + 5 \leq 0 is no solution.

Q: How do I check my work when solving an inequality?

A: To check your work when solving an inequality, you need to plug in values to ensure that the solution satisfies the inequality.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include forgetting to reverse the inequality sign when dividing or multiplying both sides of the inequality by a negative number, not checking your work by plugging in values, and not using inverse operations such as addition, subtraction, multiplication, and division.

Conclusion

Solving inequalities is an essential skill in mathematics that has numerous applications in real-life situations. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to use inverse operations, reverse the inequality sign when necessary, and check your work to ensure that the solution satisfies the inequality. With practice and patience, you can become proficient in solving inequalities and apply them to a wide range of problems.