Solve The Inequality.${ 6x^2 - 3x \ \textless \ 0 }$

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Introduction

In this article, we will delve into the world of inequalities and explore the process of solving a quadratic inequality. The given inequality is $6x^2 - 3x < 0$, and our goal is to find the values of $x$ that satisfy this inequality. We will use various techniques and strategies to solve this inequality and provide a clear understanding of the solution.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression on the left-hand side and a constant on the right-hand side. The quadratic expression is $6x^2 - 3x$, and the constant is $0$. Our goal is to find the values of $x$ that make the quadratic expression less than $0$.

Factoring the Quadratic Expression

To solve the inequality, we first need to factor the quadratic expression. We can start by factoring out the greatest common factor (GCF) of the two terms. In this case, the GCF is $3x$. Factoring out $3x$ gives us:

$$6x^2 - 3x = 3x(2x - 1)$$

Setting Up the Inequality

Now that we have factored the quadratic expression, we can set up the inequality. We want to find the values of $x$ that make the expression $3x(2x - 1)$ less than $0$. We can write this as:

$$3x(2x - 1) < 0$$

Finding the Critical Points

To solve the inequality, we need to find the critical points. The critical points are the values of $x$ that make the expression $3x(2x - 1)$ equal to $0$. We can find these points by setting each factor equal to $0$ and solving for $x$. This gives us:

$$3x = 0 \quad \text{or} \quad 2x - 1 = 0$$

Solving for $x$ gives us:

$$x = 0 \quad \text{or} \quad x = \frac{1}{2}$$

Creating a Sign Chart

To solve the inequality, we can create a sign chart. A sign chart is a table that shows the sign of the expression $3x(2x - 1)$ in different intervals. We can use the critical points to divide the number line into intervals and determine the sign of the expression in each interval.

Interval	Sign of $3x$	Sign of $(2x - 1)$	Sign of $3x(2x - 1)$
$(-\infty, 0)$	$-$	$-$	$+$
$(0, \frac{1}{2})$	$+$	$-$	$-$
$(\frac{1}{2}, \infty)$	$+$	$+$	$+$

Analyzing the Sign Chart

From the sign chart, we can see that the expression $3x(2x - 1)$ is negative in the interval $(0, \frac{1}{2})$. This means that the inequality $3x(2x - 1) < 0$ is satisfied when $0 < x < \frac{1}{2}$.

Conclusion

In this article, we solved the inequality $6x^2 - 3x < 0$ by factoring the quadratic expression, setting up the inequality, finding the critical points, creating a sign chart, and analyzing the sign chart. We found that the inequality is satisfied when $0 < x < \frac{1}{2}$. This solution provides a clear understanding of the values of $x$ that satisfy the inequality.

Final Answer

The final answer is $\boxed{(0, \frac{1}{2})}$.

Additional Tips and Tricks

• When solving a quadratic inequality, it's essential to factor the quadratic expression and set up the inequality correctly.
• Finding the critical points is a crucial step in solving a quadratic inequality.
• Creating a sign chart can help visualize the solution and make it easier to analyze.
• When analyzing the sign chart, pay attention to the intervals where the expression is negative, as these are the intervals that satisfy the inequality.

Frequently Asked Questions

• Q: What is the first step in solving a quadratic inequality? A: The first step is to factor the quadratic expression and set up the inequality correctly.
• Q: How do I find the critical points? A: To find the critical points, set each factor equal to $0$ and solve for $x$.
• Q: What is a sign chart? A: A sign chart is a table that shows the sign of the expression in different intervals.
• Q: How do I analyze the sign chart? A: Pay attention to the intervals where the expression is negative, as these are the intervals that satisfy the inequality.
  

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Introduction

In this article, we will delve into the world of inequalities and explore the process of solving a quadratic inequality. The given inequality is $6x^2 - 3x < 0$, and our goal is to find the values of $x$ that satisfy this inequality. We will use various techniques and strategies to solve this inequality and provide a clear understanding of the solution.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression on the left-hand side and a constant on the right-hand side. The quadratic expression is $6x^2 - 3x$, and the constant is $0$. Our goal is to find the values of $x$ that make the quadratic expression less than $0$.

Factoring the Quadratic Expression

To solve the inequality, we first need to factor the quadratic expression. We can start by factoring out the greatest common factor (GCF) of the two terms. In this case, the GCF is $3x$. Factoring out $3x$ gives us:

$$6x^2 - 3x = 3x(2x - 1)$$

Setting Up the Inequality

Now that we have factored the quadratic expression, we can set up the inequality. We want to find the values of $x$ that make the expression $3x(2x - 1)$ less than $0$. We can write this as:

$$3x(2x - 1) < 0$$

Finding the Critical Points

To solve the inequality, we need to find the critical points. The critical points are the values of $x$ that make the expression $3x(2x - 1)$ equal to $0$. We can find these points by setting each factor equal to $0$ and solving for $x$. This gives us:

$$3x = 0 \quad \text{or} \quad 2x - 1 = 0$$

Solving for $x$ gives us:

$$x = 0 \quad \text{or} \quad x = \frac{1}{2}$$

Creating a Sign Chart

To solve the inequality, we can create a sign chart. A sign chart is a table that shows the sign of the expression $3x(2x - 1)$ in different intervals. We can use the critical points to divide the number line into intervals and determine the sign of the expression in each interval.

Interval	Sign of $3x$	Sign of $(2x - 1)$	Sign of $3x(2x - 1)$
$(-\infty, 0)$	$-$	$-$	$+$
$(0, \frac{1}{2})$	$+$	$-$	$-$
$(\frac{1}{2}, \infty)$	$+$	$+$	$+$

Analyzing the Sign Chart

From the sign chart, we can see that the expression $3x(2x - 1)$ is negative in the interval $(0, \frac{1}{2})$. This means that the inequality $3x(2x - 1) < 0$ is satisfied when $0 < x < \frac{1}{2}$.

Conclusion

In this article, we solved the inequality $6x^2 - 3x < 0$ by factoring the quadratic expression, setting up the inequality, finding the critical points, creating a sign chart, and analyzing the sign chart. We found that the inequality is satisfied when $0 < x < \frac{1}{2}$. This solution provides a clear understanding of the values of $x$ that satisfy the inequality.

Final Answer

The final answer is $\boxed{(0, \frac{1}{2})}$.

Additional Tips and Tricks

• When solving a quadratic inequality, it's essential to factor the quadratic expression and set up the inequality correctly.
• Finding the critical points is a crucial step in solving a quadratic inequality.
• Creating a sign chart can help visualize the solution and make it easier to analyze.
• When analyzing the sign chart, pay attention to the intervals where the expression is negative, as these are the intervals that satisfy the inequality.

Frequently Asked Questions

Q&A Section

Q: What is the first step in solving a quadratic inequality?

A: The first step is to factor the quadratic expression and set up the inequality correctly.

Q: How do I find the critical points?

A: To find the critical points, set each factor equal to $0$ and solve for $x$.

Q: What is a sign chart?

A: A sign chart is a table that shows the sign of the expression in different intervals.

Q: How do I analyze the sign chart?

A: Pay attention to the intervals where the expression is negative, as these are the intervals that satisfy the inequality.

Q: What is the difference between a quadratic equation and a quadratic inequality?

A: A quadratic equation is an equation that involves a quadratic expression, while a quadratic inequality is an inequality that involves a quadratic expression.

Q: How do I determine the sign of the expression in each interval?

A: To determine the sign of the expression in each interval, use the critical points to divide the number line into intervals and determine the sign of the expression in each interval.

Q: What is the significance of the critical points?

A: The critical points are the values of $x$ that make the expression equal to $0$, and they are used to divide the number line into intervals.

Q: How do I use the sign chart to solve the inequality?

A: Use the sign chart to determine the intervals where the expression is negative, as these are the intervals that satisfy the inequality.

Q: What is the final answer to the inequality $6x^2 - 3x < 0$?

A: The final answer is $\boxed{(0, \frac{1}{2})}$.

Additional Resources

• For more information on solving quadratic inequalities, visit the Khan Academy website.
• For more practice problems on solving quadratic inequalities, visit the Mathway website.
• For more information on factoring quadratic expressions, visit the Purplemath website.

Conclusion

In this article, we solved the inequality $6x^2 - 3x < 0$ by factoring the quadratic expression, setting up the inequality, finding the critical points, creating a sign chart, and analyzing the sign chart. We found that the inequality is satisfied when $0 < x < \frac{1}{2}$. This solution provides a clear understanding of the values of $x$ that satisfy the inequality.