Solve The Inequality For X X X . 5 − 3 2 X ≥ 1 3 5-\frac{3}{2} X \geq \frac{1}{3} 5 − 2 3 ​ X ≥ 3 1 ​ A. X ≤ 28 9 X \leq \frac{28}{9} X ≤ 9 28 ​ B. X ≤ 7 X \leq 7 X ≤ 7 C. X ≥ 28 9 X \geq \frac{28}{9} X ≥ 9 28 ​ D. X ≥ 7 X \geq 7 X ≥ 7

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Introduction

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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 $5-\frac{3}{2} x \geq \frac{1}{3}$ for $x$.

Understanding the Inequality

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The given inequality is $5-\frac{3}{2} x \geq \frac{1}{3}$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to get rid of the fraction on the left-hand side by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which is 6.

Step 1: Multiply Both Sides by 6

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Multiplying both sides of the inequality by 6 gives us:

$$6\left(5-\frac{3}{2} x\right) \geq 6\left(\frac{1}{3}\right)$$

Using the distributive property, we can simplify the left-hand side of the inequality:

$$30-9x \geq 2$$

Step 2: Isolate the Variable

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Next, we need to isolate the variable $x$ on one side of the inequality sign. To do this, we can subtract 30 from both sides of the inequality:

$$-9x \geq -28$$

Step 3: Divide Both Sides by -9

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To solve for $x$, we need to divide both sides of the inequality by -9. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign:

$$x \leq \frac{28}{9}$$

Conclusion

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Therefore, the solution to the inequality $5-\frac{3}{2} x \geq \frac{1}{3}$ is $x \leq \frac{28}{9}$.

Comparison with Answer Choices

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Comparing the solution to the answer choices, we can see that the correct answer is:

• A. $x \leq \frac{28}{9}$

The other answer choices are incorrect because they do not match the solution to the inequality.

Final Answer

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The final answer is:

• A. $x \leq \frac{28}{9}$

Frequently Asked Questions

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

A: The first step in solving an inequality is to get rid of any fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: What happens when we divide or multiply an inequality by a negative number?

A: When we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign.

Q: How do we isolate the variable in an inequality?

A: To isolate the variable in an inequality, we can add or subtract the same value from both sides of the inequality, and then divide or multiply both sides of the inequality by the same non-zero value.

Additional Resources

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For more information on solving inequalities, you can check out the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Conclusion

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Solving inequalities involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, you can solve inequalities and find the solution to the inequality $5-\frac{3}{2} x \geq \frac{1}{3}$, which is $x \leq \frac{28}{9}$.

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Introduction

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In our previous article, we discussed how to solve the inequality $5-\frac{3}{2} x \geq \frac{1}{3}$ for $x$. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in solving inequalities.

Q&A

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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. This involves getting rid of any fractions, adding or subtracting the same value from both sides, and then dividing or multiplying both sides by the same non-zero value.

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

A: The first step in solving an inequality is to get rid of any fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: What happens when I divide or multiply an inequality by a negative number?

A: When you divide or multiply an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable in an inequality, you can add or subtract the same value from both sides of the inequality, and then divide or multiply both sides of the inequality by the same non-zero value.

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 $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Set each factor equal to zero and solve for $x$.
3. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \geq 0$ or $\frac{f(x)}{g(x)} \leq 0$, where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) \geq 0$ or $f(x) \leq 0$, where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

1. Factor the numerator and denominator, if possible.
2. Set each factor equal to zero and solve for $x$.
3. Use a number line or a graph to determine the intervals where the inequality is true.

Conclusion

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Solving inequalities involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, you can solve inequalities and understand the concepts and steps involved in solving inequalities.

Additional Resources

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For more information on solving inequalities, you can check out the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Frequently Asked Questions

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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 $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Set each factor equal to zero and solve for $x$.
3. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \geq 0$ or $\frac{f(x)}{g(x)} \leq 0$, where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) \geq 0$ or $f(x) \leq 0$, where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

1. Factor the numerator and denominator, if possible.
2. Set each factor equal to zero and solve for $x$.
3. Use a number line or a graph to determine the intervals where the inequality is true.

Final Answer

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The final answer is that solving inequalities involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, you can solve inequalities and understand the concepts and steps involved in solving inequalities.