QuestionWhich Of The Following Values Are Solutions To The Inequality 2 X − 8 \textless 2 2x - 8 \ \textless \ 2 2 X − 8 \textless 2 ?I. 3 3 3 II. 6 6 6 III. 5 5 5 Answer Choices:A. None B. II Only C. I Only D. I And II E. III Only F. II And III G. I

Understanding the Inequality

In this article, we will explore the concept of solving inequalities and apply it to a specific problem. The given inequality is $2x - 8 \ \textless \ 2$. Our goal is to find the values of $x$ that satisfy this inequality.

Step 1: Add 8 to Both Sides

To solve the inequality, we start by adding 8 to both sides of the equation. This gives us:

$$2x - 8 + 8 \ \textless \ 2 + 8$$

Simplifying the equation, we get:

$$2x \ \textless \ 10$$

Step 2: Divide Both Sides by 2

Next, we divide both sides of the inequality by 2 to isolate $x$. This gives us:

$$\frac{2x}{2} \ \textless \ \frac{10}{2}$$

Simplifying the equation, we get:

$$x \ \textless \ 5$$

Analyzing the Solutions

Now that we have solved the inequality, we need to analyze the solutions. The inequality $x \ \textless \ 5$ means that $x$ is less than 5. In other words, any value of $x$ that is less than 5 is a solution to the inequality.

Evaluating the Answer Choices

Let's evaluate the answer choices based on our analysis:

• I. $3$: Since $3$ is less than 5, it is a solution to the inequality.
• II. $6$: Since $6$ is not less than 5, it is not a solution to the inequality.
• III. $5$: Since $5$ is not less than 5, it is not a solution to the inequality.

Conclusion

Based on our analysis, the correct answer is:

• C. I only

This means that only $3$ is a solution to the inequality $2x - 8 \ \textless \ 2$.

Additional Tips and Tricks

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

• When adding or subtracting a value to both sides of an inequality, the direction of the inequality sign remains the same.
• When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality sign is reversed.
• When solving inequalities, it's crucial to analyze the solutions and determine which values satisfy the inequality.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations when solving inequalities.
• Not analyzing the solutions and determining which values satisfy the inequality.
• Not considering the direction of the inequality sign when adding or subtracting values.

Conclusion

In conclusion, solving inequalities requires a step-by-step approach and attention to detail. By following the tips and tricks outlined in this article, you can confidently solve inequalities and determine which values satisfy the inequality. Remember to analyze the solutions and avoid common mistakes to ensure accurate results.

Final Thoughts

Solving inequalities is a fundamental concept in mathematics that has numerous applications in real-world scenarios. By mastering the art of solving inequalities, you can tackle complex problems and make informed decisions. Remember to practice regularly and seek help when needed to become proficient in solving inequalities.

Additional Resources

For further learning and practice, here are some additional resources:

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

Conclusion

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Q: What is an inequality?

A: An inequality is a statement that compares two values using a mathematical symbol, such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable (usually x) on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that says two values are equal, while an inequality is a statement that compares two values using a mathematical symbol.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. This will not change the direction of the inequality sign.

Q: Can I multiply or divide both sides of an inequality by a negative value?

A: No, you cannot multiply or divide both sides of an inequality by a negative value. This will change the direction of the inequality sign.

Q: How do I determine which values satisfy an inequality?

A: To determine which values satisfy an inequality, you need to analyze the solutions and determine which values are greater than, less than, or equal to the value on the other side of the inequality sign.

Q: What is the order of operations when solving inequalities?

A: The order of operations when solving inequalities is the same as when solving equations:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, make sure to check your work and verify that the solution is correct.

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

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the value on the other side of the inequality sign. Then, shade the region to the left or right of the point, depending on the direction of the inequality sign.

Q: Can I solve a system of inequalities?

A: Yes, you can solve a system of inequalities by finding the intersection of the solution sets of each inequality.

Q: How do I determine which values satisfy a system of inequalities?

A: To determine which values satisfy a system of inequalities, you need to find the intersection of the solution sets of each inequality.

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

A: A linear inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants. A nonlinear inequality is an inequality that cannot be written in this form.

Q: Can I solve a nonlinear inequality?

A: Yes, you can solve a nonlinear inequality by using algebraic techniques, such as factoring or using the quadratic formula.

Conclusion

In this article, we answered frequently asked questions about solving inequalities. We covered topics such as the difference between an inequality and an equation, the order of operations, and how to graph an inequality on a number line. We also discussed how to solve a system of inequalities and the difference between a linear inequality and a nonlinear inequality. Remember to practice regularly and seek help when needed to become proficient in solving inequalities.

Additional Resources

For further learning and practice, here are some additional resources:

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

Final Thoughts

Solving inequalities is a fundamental concept in mathematics that has numerous applications in real-world scenarios. By mastering the art of solving inequalities, you can tackle complex problems and make informed decisions. Remember to practice regularly and seek help when needed to become proficient in solving inequalities.