Solve The Inequality: 3 X − X 2 \textgreater 1 \frac{3}{x} - \frac{x}{2} \ \textgreater \ 1 X 3 ​ − 2 X ​ \textgreater 1

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater than or less than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $\frac{3}{x} - \frac{x}{2} \ \textgreater \ 1$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Move all terms to the left-hand side

To solve the inequality, we need to isolate the variable $x$ on one side of the inequality. The first step is to move all terms to the left-hand side of the inequality. We can do this by subtracting 1 from both sides of the inequality.

$\frac{3}{x} - \frac{x}{2} - 1 \ \textgreater \ 0$

Step 2: Find a common denominator

The next step is to find a common denominator for the fractions. The common denominator is the least common multiple of the denominators of the fractions. In this case, the denominators are $x$ and $2$. The least common multiple of $x$ and $2$ is $2x$.

$\frac{3}{x} - \frac{x}{2} - \frac{2}{2x} \ \textgreater \ 0$

Step 3: Simplify the inequality

Now that we have a common denominator, we can simplify the inequality by combining the fractions.

$\frac{6 - x^2 - 2}{2x} \ \textgreater \ 0$

Step 4: Factor the numerator

The next step is to factor the numerator of the fraction. We can factor the numerator as follows:

$\frac{(2 - x)(3 + x)}{2x} \ \textgreater \ 0$

Step 5: Determine the sign of the inequality

Now that we have factored the numerator, we can determine the sign of the inequality. The sign of the inequality will depend on the signs of the factors in the numerator and the denominator.

Case 1: $x > 0$

If $x > 0$, then the denominator is positive, and the sign of the inequality will depend on the signs of the factors in the numerator.

• If $2 - x > 0$ and $3 + x > 0$, then the inequality is true.
• If $2 - x < 0$ and $3 + x > 0$, then the inequality is false.
• If $2 - x > 0$ and $3 + x < 0$, then the inequality is false.
• If $2 - x < 0$ and $3 + x < 0$, then the inequality is true.

Case 2: $x < 0$

If $x < 0$, then the denominator is negative, and the sign of the inequality will depend on the signs of the factors in the numerator.

• If $2 - x > 0$ and $3 + x > 0$, then the inequality is false.
• If $2 - x < 0$ and $3 + x > 0$, then the inequality is true.
• If $2 - x > 0$ and $3 + x < 0$, then the inequality is true.
• If $2 - x < 0$ and $3 + x < 0$, then the inequality is false.

Conclusion

In conclusion, the solution to the inequality $\frac{3}{x} - \frac{x}{2} \ \textgreater \ 1$ is $x \in (-\infty, -3) \cup (0, 2)$. This means that the inequality is true for all values of $x$ less than -3 and all values of $x$ greater than 0 and less than 2.

Graphical Representation

The solution to the inequality can be represented graphically as follows:

• The inequality is true for all values of $x$ less than -3.
• The inequality is true for all values of $x$ greater than 0 and less than 2.

Final Answer

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Introduction

In our previous article, we solved the inequality $\frac{3}{x} - \frac{x}{2} \ \textgreater \ 1$ using a step-by-step approach. In this article, we will answer some common questions that readers may have about solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often with a greater than or less than symbol.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing 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 $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

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

Q: What is the significance of the sign of the factors in solving inequalities?

A: The sign of the factors in solving inequalities is crucial in determining the solution. If the factors are positive, the inequality is true. If the factors are negative, the inequality is false.

Q: How do I graph the solution to an inequality?

A: To graph the solution to an inequality, you need to plot the points on the number line that satisfy the inequality. You can use a number line to graph the solution to an inequality.

Q: What is the difference between a solution set and a graph?

A: A solution set is a set of values that satisfy the inequality, while a graph is a visual representation of the solution set.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug in values from the solution set into the original inequality and check if the inequality is true.

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

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

• Not isolating the variable on one side of the inequality
• Not using the correct sign of the factors
• Not checking the solution set
• Not graphing the solution set

Conclusion

In conclusion, solving inequalities requires a step-by-step approach and attention to detail. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to check your solution and graph the solution set to ensure that you have found the correct solution.

Final Answer

The final answer is $\boxed{(-\infty, -3) \cup (0, 2)}$.