Inequalities and Quadratic Equations

Oct 13, 2025 by miodeya.com 57 views

Introduction

When dealing with real-world problems, such as building construction or design, it's essential to understand the mathematical concepts that govern these situations. One common problem is finding the height of a wall given the combined area. In this article, we'll explore the inequality that can be used to solve for the height of the wall, and we'll examine the different options provided.

Understanding the Problem

The problem involves finding the height of a wall, denoted as $x$ , given the combined area. This means we need to find the value of $x$ that satisfies the given condition. To do this, we'll use inequalities, which are mathematical expressions that describe a range of values.

Inequalities and Quadratic Equations

Inequalities involving quadratic equations are a common way to describe the relationship between variables. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this case, we're dealing with quadratic inequalities, which are inequalities that involve quadratic expressions.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$ , the solutions are given by:

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

However, when dealing with inequalities, we need to consider the sign of the expression inside the square root. If the expression is positive, the inequality is satisfied when the expression is greater than or equal to zero. If the expression is negative, the inequality is satisfied when the expression is less than or equal to zero.

Analyzing the Options

Let's analyze the options provided:

Option A: $x^2 + 15x - 166 \leq 0$

To determine if this inequality is satisfied, we need to find the values of $x$ that make the expression inside the square root positive. We can do this by factoring the quadratic expression:

x^2 + 15x - 166 = (x + 22)(x - 7)

Setting each factor equal to zero, we get:

x + 22 = 0 \quad \text{or} \quad x - 7 = 0

Solving for $x$ , we get:

x = -22 \quad \text{or} \quad x = 7

Since the inequality is satisfied when the expression is less than or equal to zero, we need to find the values of $x$ that make the expression negative. This occurs when $-22 < x < 7$ .

Option B: $x^2 + 15x + 166 \leq 0$

To determine if this inequality is satisfied, we need to find the values of $x$ that make the expression inside the square root positive. We can do this by factoring the quadratic expression:

x^2 + 15x + 166 = (x + 13)(x + 13)

Setting each factor equal to zero, we get:

x + 13 = 0 \quad \text{or} \quad x + 13 = 0

Solving for $x$ , we get:

x = -13

Since the inequality is satisfied when the expression is less than or equal to zero, we need to find the values of $x$ that make the expression negative. This occurs when $x = -13$ .

Option C: $x^2 + 15x + 238 \leq 0$

To determine if this inequality is satisfied, we need to find the values of $x$ that make the expression inside the square root positive. We can do this by factoring the quadratic expression:

x^2 + 15x + 238 = (x + 14)(x + 17)

Setting each factor equal to zero, we get:

x + 14 = 0 \quad \text{or} \quad x + 17 = 0

Solving for $x$ , we get:

x = -14 \quad \text{or} \quad x = -17

Since the inequality is satisfied when the expression is less than or equal to zero, we need to find the values of $x$ that make the expression negative. This occurs when $-17 < x < -14$ .

Option D: $x^2 - 190 \leq 0$

To determine if this inequality is satisfied, we need to find the values of $x$ that make the expression inside the square root positive. We can do this by factoring the quadratic expression:

x^2 - 190 = (x - \sqrt{190})(x + \sqrt{190})

Setting each factor equal to zero, we get:

x - \sqrt{190} = 0 \quad \text{or} \quad x + \sqrt{190} = 0

Solving for $x$ , we get:

x = \sqrt{190} \quad \text{or} \quad x = -\sqrt{190}

Since the inequality is satisfied when the expression is less than or equal to zero, we need to find the values of $x$ that make the expression negative. This occurs when $-\sqrt{190} < x < \sqrt{190}$ .

Conclusion

In conclusion, the inequality that can be used to solve for the height of the wall is:

x^2 + 15x - 166 \leq 0

This inequality is satisfied when $-22 < x < 7$ . The other options do not satisfy the condition, and therefore, cannot be used to solve for the height of the wall.

Final Answer

The final answer is: $\boxed{A}$

Introduction

In our previous article, we explored the inequality that can be used to solve for the height of a wall. We analyzed the different options provided and determined that the correct inequality is:

x^2 + 15x - 166 \leq 0

This inequality is satisfied when $-22 < x < 7$ . In this article, we'll provide a Q&A section to help clarify any doubts and provide additional information.

Q&A

Q: What is the significance of the inequality $x^2 + 15x - 166 \leq 0$ ?

A: The inequality $x^2 + 15x - 166 \leq 0$ represents the range of values for the height of the wall. The inequality is satisfied when the expression inside the square root is less than or equal to zero.

Q: How do I determine the values of $x$ that satisfy the inequality?

A: To determine the values of $x$ that satisfy the inequality, you need to find the values of $x$ that make the expression inside the square root positive. You can do this by factoring the quadratic expression:

x^2 + 15x - 166 = (x + 22)(x - 7)

Setting each factor equal to zero, you get:

x + 22 = 0 \quad \text{or} \quad x - 7 = 0

Solving for $x$ , you get:

x = -22 \quad \text{or} \quad x = 7

Since the inequality is satisfied when the expression is less than or equal to zero, you need to find the values of $x$ that make the expression negative. This occurs when $-22 < x < 7$ .

Q: What is the relationship between the inequality and the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. However, when dealing with inequalities, you need to consider the sign of the expression inside the square root. If the expression is positive, the inequality is satisfied when the expression is greater than or equal to zero. If the expression is negative, the inequality is satisfied when the expression is less than or equal to zero.

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

A: No, you cannot use the quadratic formula to solve the inequality. The quadratic formula is used to solve quadratic equations, not inequalities. To solve the inequality, you need to use the method of factoring or the method of completing the square.

Q: What is the significance of the values $-22$ and $7$ in the inequality?

A: The values $-22$ and $7$ represent the critical points of the inequality. These values are obtained by setting each factor of the quadratic expression equal to zero. The inequality is satisfied when the expression is less than or equal to zero, which occurs when $-22 < x < 7$ .

Q: Can I use the inequality to solve for the height of a wall with a different shape?

A: No, the inequality is specific to the shape of the wall. If the shape of the wall is different, you will need to use a different inequality to solve for the height.

Conclusion

In conclusion, the inequality $x^2 + 15x - 166 \leq 0$ is used to solve for the height of a wall. The inequality is satisfied when $-22 < x < 7$ . We hope this Q&A section has helped clarify any doubts and provided additional information.

Final Answer

The final answer is: $\boxed{A}$