Which Of The Following Values Are Solutions To The Inequality 6 \textless 4 X + 9 6 \ \textless \ 4x + 9 6 \textless 4 X + 9 ?I. 2 II. 6 III. -2 A. None B. I Only C. II Only D. III Only E. I And II F. I And III G. II And III H. I, II, And III

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Understanding the Inequality

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To solve the inequality $6 \ \textless \ 4x + 9$, we need to isolate the variable $x$ on one side of the inequality. This will give us the range of values for which the inequality holds true.

Step 1: Subtract 9 from Both Sides

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We start by subtracting 9 from both sides of the inequality:

$$6 \ \textless \ 4x + 9$$

Subtracting 9 from both sides gives us:

$$-3 \ \textless \ 4x$$

Step 2: Divide Both Sides by 4

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Next, we divide both sides of the inequality by 4:

$$-3 \ \textless \ 4x$$

Dividing both sides by 4 gives us:

$$-\frac{3}{4} \ \textless \ x$$

Analyzing the Solutions

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Now that we have the inequality in the form $x \ \textless \ -\frac{3}{4}$, we can analyze the solutions.

What are the Solutions?

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The solutions to the inequality are all values of $x$ that satisfy the inequality. In this case, the solutions are all values of $x$ that are less than $-\frac{3}{4}$.

Checking the Answer Choices

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Now, let's check the answer choices to see which ones are solutions to the inequality.

I. 2

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Is 2 a solution to the inequality $x \ \textless \ -\frac{3}{4}$? No, 2 is not less than $-\frac{3}{4}$, so it is not a solution.

II. 6

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Is 6 a solution to the inequality $x \ \textless \ -\frac{3}{4}$? No, 6 is not less than $-\frac{3}{4}$, so it is not a solution.

III. -2

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Is -2 a solution to the inequality $x \ \textless \ -\frac{3}{4}$? Yes, -2 is less than $-\frac{3}{4}$, so it is a solution.

Conclusion

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Based on our analysis, the only solution to the inequality $6 \ \textless \ 4x + 9$ is III. -2.

The final answer is D. III only.

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Frequently Asked Questions

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Q: What is the first step in solving the inequality $6 \ \textless \ 4x + 9$?

A: The first step in solving the inequality is to subtract 9 from both sides, which gives us $-3 \ \textless \ 4x$.

Q: Why do we need to isolate the variable $x$ on one side of the inequality?

A: We need to isolate the variable $x$ on one side of the inequality in order to find the range of values for which the inequality holds true.

Q: What is the next step after subtracting 9 from both sides?

A: The next step is to divide both sides of the inequality by 4, which gives us $-\frac{3}{4} \ \textless \ x$.

Q: What are the solutions to the inequality $x \ \textless \ -\frac{3}{4}$?

A: The solutions to the inequality are all values of $x$ that are less than $-\frac{3}{4}$.

Q: Which of the answer choices are solutions to the inequality $6 \ \textless \ 4x + 9$?

A: Based on our analysis, the only solution to the inequality is III. -2.

Q: Why is 2 not a solution to the inequality $x \ \textless \ -\frac{3}{4}$?

A: 2 is not a solution to the inequality because it is not less than $-\frac{3}{4}$.

Q: Why is 6 not a solution to the inequality $x \ \textless \ -\frac{3}{4}$?

A: 6 is not a solution to the inequality because it is not less than $-\frac{3}{4}$.

Q: Why is -2 a solution to the inequality $x \ \textless \ -\frac{3}{4}$?

A: -2 is a solution to the inequality because it is less than $-\frac{3}{4}$.

Additional Tips and Tricks

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Tip 1: Make sure to isolate the variable on one side of the inequality.

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When solving an inequality, it's essential to isolate the variable on one side of the inequality. This will give you the range of values for which the inequality holds true.

Tip 2: Check your answer choices carefully.

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When checking your answer choices, make sure to carefully evaluate each option to see if it satisfies the inequality.

Tip 3: Use a number line to visualize the solutions.

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Using a number line can help you visualize the solutions to the inequality and make it easier to identify the correct answer.

Conclusion

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Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to isolate the variable on one side of the inequality, check your answer choices carefully, and use a number line to visualize the solutions.

The final answer is D. III only.