What Is The First Step In Solving The Inequality $\frac{m-2}{6}\ \textless \ -1$?A. Multiply Both Sides By 6.B. Add 2 To Both Sides.C. Change The Direction Of The Inequality.D. Change The Inequality To $\leq$.

$$

Understanding the Basics of Inequality Solving

When dealing with inequalities, it's essential to understand the basic steps involved in solving them. Inequalities are mathematical expressions that compare two values, and solving them requires a systematic approach. The first step in solving an inequality is crucial, as it sets the stage for the rest of the solution process.

The Importance of the First Step

The first step in solving an inequality is to isolate the variable on one side of the inequality sign. This can be achieved by performing operations on both sides of the inequality, such as multiplying or dividing by a constant, or adding or subtracting a value. However, it's essential to remember that when multiplying or dividing both sides of an inequality by a constant, the direction of the inequality sign may change.

Analyzing the Given Inequality

The given inequality is $\frac{m-2}{6}\ \textless \ -1$. To solve this inequality, we need to isolate the variable $m$ on one side of the inequality sign. The first step in solving this inequality is to eliminate the fraction by multiplying both sides by the denominator, which is 6.

Eliminating the Fraction

To eliminate the fraction, we multiply both sides of the inequality by 6. This operation is valid because 6 is a non-zero constant, and multiplying both sides of an inequality by a non-zero constant does not change the direction of the inequality sign.

The Correct Answer

The correct answer is A. Multiply both sides by 6. This is the first step in solving the inequality $\frac{m-2}{6}\ \textless \ -1$.

Why the Other Options are Incorrect

The other options are incorrect because they do not address the first step in solving the inequality. Option B, Add 2 to both sides, is incorrect because it does not eliminate the fraction. Option C, Change the direction of the inequality, is incorrect because it is not necessary to change the direction of the inequality sign at this stage. Option D, Change the inequality to $\leq$, is incorrect because it is not relevant to the first step in solving the inequality.

Conclusion

In conclusion, the first step in solving the inequality $\frac{m-2}{6}\ \textless \ -1$ is to multiply both sides by 6. This operation eliminates the fraction and sets the stage for the rest of the solution process.

Additional Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always check the direction of the inequality sign after multiplying or dividing both sides by a constant.
• Use the correct operation to eliminate the fraction, such as multiplying or dividing both sides by the denominator.
• Be careful when adding or subtracting values to both sides of the inequality, as this can change the direction of the inequality sign.

Real-World Applications

Inequalities have numerous real-world applications, such as:

• Modeling population growth and decline
• Analyzing financial data and making investment decisions
• Solving optimization problems in business and economics
• Understanding and predicting weather patterns

Final Thoughts

In conclusion, solving inequalities requires a systematic approach, and the first step is crucial in setting the stage for the rest of the solution process. By following the correct steps and using the right operations, you can solve inequalities with confidence and apply them to real-world problems.

Frequently Asked Questions

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality sign.

Q: How do I eliminate a fraction in an inequality?

A: To eliminate a fraction, multiply both sides of the inequality by the denominator.

Q: What happens when I multiply or divide both sides of an inequality by a constant?

A: When you multiply or divide both sides of an inequality by a constant, the direction of the inequality sign may change.

Q: What are some real-world applications of inequalities?

A: Inequalities have numerous real-world applications, such as modeling population growth and decline, analyzing financial data, and solving optimization problems in business and economics.

References

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Math Open Reference
• [3] "Inequalities in Real-World Applications" by Wolfram Alpha

Further Reading

• "Inequalities and Their Applications" by Springer
• "Solving Inequalities: A Comprehensive Guide" by CRC Press
• "Inequalities in Mathematics and Science" by Cambridge University Press
  

Q&A: Solving Inequalities

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality sign. This can be achieved by performing operations on both sides of the inequality, such as multiplying or dividing by a constant, or adding or subtracting a value.

Q: How do I eliminate a fraction in an inequality?

A: To eliminate a fraction, multiply both sides of the inequality by the denominator. This operation is valid because the denominator is a non-zero constant, and multiplying both sides of an inequality by a non-zero constant does not change the direction of the inequality sign.

Q: What happens when I multiply or divide both sides of an inequality by a constant?

A: When you multiply or divide both sides of an inequality by a constant, the direction of the inequality sign may change. If the constant is positive, the direction of the inequality sign remains the same. However, if the constant is negative, the direction of the inequality sign is reversed.

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

A: Some common mistakes to avoid when solving inequalities include:

• Not checking the direction of the inequality sign after multiplying or dividing both sides by a constant.
• Not using the correct operation to eliminate the fraction, such as multiplying or dividing both sides by the denominator.
• Not being careful when adding or subtracting values to both sides of the inequality, as this can change the direction of the inequality sign.

Q: How do I determine the direction of the inequality sign?

A: To determine the direction of the inequality sign, you can use the following rules:

• If the constant is positive, the direction of the inequality sign remains the same.
• If the constant is negative, the direction of the inequality sign is reversed.
• If the inequality is of the form $a \leq b$, the direction of the inequality sign is always non-strict (i.e., $\leq$).

Q: What are some real-world applications of inequalities?

A: Inequalities have numerous real-world applications, such as:

• Modeling population growth and decline
• Analyzing financial data and making investment decisions
• Solving optimization problems in business and economics
• Understanding and predicting weather patterns

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you can use the following steps:

• Graph the inequalities on a coordinate plane.
• Find the intersection of the two inequalities.
• Check the intersection point to determine if it satisfies both inequalities.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: inequalities of the form $ax + by \leq c$ or $ax + by \geq c$.
• Quadratic inequalities: inequalities of the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$.
• Rational inequalities: inequalities of the form $\frac{ax + b}{cx + d} \leq 0$ or $\frac{ax + b}{cx + d} \geq 0$.

Q: How do I graph an inequality on a coordinate plane?

A: To graph an inequality on a coordinate plane, you can use the following steps:

• Plot the boundary line of the inequality.
• Test a point on either side of the boundary line to determine the direction of the inequality sign.
• Shade the region that satisfies the inequality.

Additional Resources

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Math Open Reference
• [3] "Inequalities in Real-World Applications" by Wolfram Alpha

Further Reading

• "Inequalities and Their Applications" by Springer
• "Solving Inequalities: A Comprehensive Guide" by CRC Press
• "Inequalities in Mathematics and Science" by Cambridge University Press