Find The Set Of Possible Values Of $x$ For Which:1. $4x^2 - 9 \ \textless \ 0$2. $ 8 − 6 X − 5 X 2 \textgreater 0 8 - 6x - 5x^2 \ \textgreater \ 0 8 − 6 X − 5 X 2 \textgreater 0 [/tex]You Must Show All Your Working.$ \begin{array}{l} 4x^2 - 9 \ \textless \ 0 \ 4x^2

Introduction

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 finding the set of possible values of the variable that satisfy the given inequality. In this article, we will focus on solving two quadratic inequalities: $4x^2 - 9 < 0$ and $8 - 6x - 5x^2 > 0$. We will use algebraic techniques to find the possible values of x that satisfy each inequality.

Solving the First Inequality: $4x^2 - 9 < 0$

To solve the inequality $4x^2 - 9 < 0$, we need to find the values of x that make the expression $4x^2 - 9$ negative. We can start by adding 9 to both sides of the inequality, which gives us:

$$4x^2 < 9$$

Next, we can divide both sides of the inequality by 4, which gives us:

$$x^2 < \frac{9}{4}$$

Now, we can take the square root of both sides of the inequality. Since we are dealing with a quadratic inequality, we need to consider both the positive and negative square roots. This gives us:

$$|x| < \frac{3}{2}$$

The absolute value inequality $|x| < \frac{3}{2}$ can be rewritten as a double inequality:

$$-\frac{3}{2} < x < \frac{3}{2}$$

Therefore, the possible values of x that satisfy the inequality $4x^2 - 9 < 0$ are all real numbers between -3/2 and 3/2.

Solving the Second Inequality: $8 - 6x - 5x^2 > 0$

To solve the inequality $8 - 6x - 5x^2 > 0$, we need to find the values of x that make the expression $8 - 6x - 5x^2$ positive. We can start by rearranging the inequality to get:

$$-5x^2 - 6x + 8 > 0$$

Next, we can multiply both sides of the inequality by -1, which gives us:

$$5x^2 + 6x - 8 < 0$$

Now, we can factor the quadratic expression on the left-hand side of the inequality:

$$(5x - 4)(x + 2) < 0$$

To find the values of x that satisfy the inequality, we need to consider the signs of the two factors $(5x - 4)$ and $(x + 2)$. We can create a sign chart to help us determine the intervals where the inequality is satisfied:

Interval	$(5x - 4)$	$(x + 2)$	$(5x - 4)(x + 2)$
$x < -2$	-	-	+
$-2 < x < 4/5$	-	+	-
$x > 4/5$	+	+	+

From the sign chart, we can see that the inequality $(5x - 4)(x + 2) < 0$ is satisfied when $-2 < x < 4/5$. Therefore, the possible values of x that satisfy the inequality $8 - 6x - 5x^2 > 0$ are all real numbers between -2 and 4/5.

Conclusion

In this article, we solved two quadratic inequalities: $4x^2 - 9 < 0$ and $8 - 6x - 5x^2 > 0$. We used algebraic techniques to find the possible values of x that satisfy each inequality. For the first inequality, we found that the possible values of x are all real numbers between -3/2 and 3/2. For the second inequality, we found that the possible values of x are all real numbers between -2 and 4/5. We hope that this article has provided a clear and concise explanation of how to solve quadratic inequalities.

References

• [1] Larson, R. (2014). College Algebra. Cengage Learning.
• [2] Rogawski, J. (2011). Calculus. W.H. Freeman and Company.
• [3] Stewart, J. (2011). Calculus. Cengage Learning.

Keywords

• Quadratic inequalities
• Algebraic techniques
• Inequality solutions
• Possible values of x
• Real numbers
• Double inequality
• Absolute value inequality
• Sign chart
• Quadratic expression
• Factoring
• Algebraic manipulation
  

Introduction

Quadratic 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 quadratic inequalities involves finding the set of possible values of the variable that satisfy the given inequality. In this article, we will provide a Q&A guide to help you understand how to solve quadratic inequalities.

Q: What is a quadratic inequality?

A: A quadratic 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, where the variable is raised to the power of 2.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to follow these steps:

1. Rearrange the inequality to get the quadratic expression on one side of the inequality sign.
2. Factor the quadratic expression, if possible.
3. Create a sign chart to determine the intervals where the inequality is satisfied.
4. Identify the possible values of x that satisfy the inequality.

Q: What is a sign chart?

A: A sign chart is a table that shows the signs of the factors of the quadratic expression in different intervals. It helps you determine the intervals where the inequality is satisfied.

Q: How do I create a sign chart?

A: To create a sign chart, you need to:

1. Identify the critical points of the quadratic expression, which are the values of x that make the quadratic expression equal to zero.
2. Create a table with the intervals between the critical points.
3. Determine the signs of the factors of the quadratic expression in each interval.
4. Use the signs of the factors to determine the intervals where the inequality is satisfied.

Q: What are the critical points of a quadratic expression?

A: The critical points of a quadratic expression are the values of x that make the quadratic expression equal to zero. They are the roots of the quadratic equation.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the roots of a quadratic equation. It is used to solve quadratic equations of the form ax^2 + bx + c = 0.

Q: Can I use the quadratic formula to solve quadratic inequalities?

A: No, the quadratic formula is used to solve quadratic equations, not quadratic inequalities. To solve quadratic inequalities, you need to use the steps outlined above.

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

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

• Not rearranging the inequality to get the quadratic expression on one side of the inequality sign.
• Not factoring the quadratic expression, if possible.
• Not creating a sign chart to determine the intervals where the inequality is satisfied.
• Not identifying the possible values of x that satisfy the inequality.

Q: How do I check my solution to a quadratic inequality?

A: To check your solution to a quadratic inequality, you need to:

1. Plug in a value of x from each interval into the original inequality.
2. Check if the inequality is satisfied for each value of x.
3. If the inequality is satisfied for all values of x in the interval, then the solution is correct.

Conclusion

Quadratic 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 quadratic inequalities involves finding the set of possible values of the variable that satisfy the given inequality. In this article, we provided a Q&A guide to help you understand how to solve quadratic inequalities. We hope that this article has provided a clear and concise explanation of how to solve quadratic inequalities.

References

• [1] Larson, R. (2014). College Algebra. Cengage Learning.
• [2] Rogawski, J. (2011). Calculus. W.H. Freeman and Company.
• [3] Stewart, J. (2011). Calculus. Cengage Learning.

Keywords

• Quadratic inequalities
• Algebraic techniques
• Inequality solutions
• Possible values of x
• Real numbers
• Double inequality
• Absolute value inequality
• Sign chart
• Quadratic expression
• Factoring
• Algebraic manipulation
• Quadratic formula
• Roots of a quadratic equation
• Critical points
• Intervals
• Inequality satisfaction