Which Is The Solution To The Inequality 5 X 2 + 13 X ≥ − 6 5x^2 + 13x \geq -6 5 X 2 + 13 X ≥ − 6 ?A. X ≤ − 2 X \leq -2 X ≤ − 2 Or X ≥ − 3 5 X \geq -\frac{3}{5} X ≥ − 5 3 ​ B. − 2 ≤ X ≤ − 3 5 -2 \leq X \leq -\frac{3}{5} − 2 ≤ X ≤ − 5 3 ​ C. X \textless − 2 X \ \textless \ -2 X \textless − 2 Or $x \ \textgreater \

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $5x^2 + 13x \geq -6$ and explore the different methods and techniques used to find the solution.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression on one side of the inequality sign. The quadratic expression is $5x^2 + 13x$, and the constant term is $-6$. To solve this inequality, we need to find the values of $x$ that make the quadratic expression greater than or equal to $-6$.

Factoring the Quadratic Expression

One of the methods to solve quadratic inequalities is to factor the quadratic expression. However, in this case, the quadratic expression $5x^2 + 13x$ cannot be factored easily. Therefore, we will use the method of completing the square to solve the inequality.

Completing the Square

To complete the square, we need to rewrite the quadratic expression in the form $(x + a)^2 + b$. We can start by dividing the quadratic expression by $5$, which gives us $x^2 + \frac{13}{5}x \geq -\frac{6}{5}$. Now, we need to add and subtract the square of half of the coefficient of $x$ to complete the square.

Adding and Subtracting the Square of Half of the Coefficient of $x$

The coefficient of $x$ is $\frac{13}{5}$. Half of this coefficient is $\frac{13}{10}$. The square of half of the coefficient of $x$ is $\left(\frac{13}{10}\right)^2 = \frac{169}{100}$. We add and subtract this value to the quadratic expression to complete the square.

Rewriting the Quadratic Expression

After adding and subtracting the square of half of the coefficient of $x$, we get:

$$x^2 + \frac{13}{5}x + \frac{169}{100} - \frac{169}{100} \geq -\frac{6}{5}$$

Simplifying the Quadratic Expression

We can simplify the quadratic expression by combining the like terms:

$$\left(x + \frac{13}{10}\right)^2 - \frac{169}{100} \geq -\frac{6}{5}$$

Adding $\frac{169}{100}$ to Both Sides

To isolate the squared term, we add $\frac{169}{100}$ to both sides of the inequality:

$$\left(x + \frac{13}{10}\right)^2 \geq -\frac{6}{5} + \frac{169}{100}$$

Simplifying the Right-Hand Side

We can simplify the right-hand side of the inequality by finding a common denominator:

$$\left(x + \frac{13}{10}\right)^2 \geq \frac{-120 + 169}{100}$$

Simplifying the Right-Hand Side (Continued)

We can simplify the right-hand side of the inequality further:

$$\left(x + \frac{13}{10}\right)^2 \geq \frac{49}{100}$$

Taking the Square Root of Both Sides

To solve the inequality, we take the square root of both sides:

$$\left|x + \frac{13}{10}\right| \geq \frac{7}{10}$$

Solving the Absolute Value Inequality

To solve the absolute value inequality, we need to consider two cases:

Case 1: $x + \frac{13}{10} \geq \frac{7}{10}$

Case 2: $x + \frac{13}{10} \leq -\frac{7}{10}$

Solving Case 1

For Case 1, we can subtract $\frac{13}{10}$ from both sides of the inequality:

$$x \geq -\frac{13}{10} + \frac{7}{10}$$

Simplifying the Right-Hand Side

We can simplify the right-hand side of the inequality:

$$x \geq -\frac{6}{10}$$

Simplifying the Right-Hand Side (Continued)

We can simplify the right-hand side of the inequality further:

$$x \geq -\frac{3}{5}$$

Solving Case 2

For Case 2, we can subtract $\frac{13}{10}$ from both sides of the inequality:

$$x \leq -\frac{13}{10} - \frac{7}{10}$$

Simplifying the Right-Hand Side

We can simplify the right-hand side of the inequality:

$$x \leq -\frac{20}{10}$$

Simplifying the Right-Hand Side (Continued)

We can simplify the right-hand side of the inequality further:

$$x \leq -2$$

Combining the Solutions

The solution to the inequality is the combination of the solutions to Case 1 and Case 2:

$$x \leq -2 \text{ or } x \geq -\frac{3}{5}$$

Conclusion

In this article, we solved the quadratic inequality $5x^2 + 13x \geq -6$ using the method of completing the square. We found that the solution to the inequality is $x \leq -2 \text{ or } x \geq -\frac{3}{5}$. This solution can be represented graphically as a shaded region on a number line, where the values of $x$ that satisfy the inequality are indicated by the shaded region.

Final Answer

The final answer to the inequality $5x^2 + 13x \geq -6$ is:

• $x \leq -2 \text{ or } x \geq -\frac{3}{5}$
  

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Introduction

In our previous article, we solved the quadratic inequality $5x^2 + 13x \geq -6$ using the method of completing the square. In this article, we will provide a Q&A guide to help you understand the solution to the inequality.

Q: What is the solution to the inequality $5x^2 + 13x \geq -6$?

A: The solution to the inequality is $x \leq -2 \text{ or } x \geq -\frac{3}{5}$.

Q: How do I represent the solution graphically?

A: You can represent the solution graphically as a shaded region on a number line, where the values of $x$ that satisfy the inequality are indicated by the shaded region.

Q: What is the significance of the absolute value in the solution?

A: The absolute value in the solution indicates that the values of $x$ that satisfy the inequality are either less than or equal to $-2$ or greater than or equal to $-\frac{3}{5}$.

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 consider two cases:

Case 1: $x + \frac{13}{10} \geq \frac{7}{10}$

Case 2: $x + \frac{13}{10} \leq -\frac{7}{10}$

Q: What is the difference between Case 1 and Case 2?

A: In Case 1, the values of $x$ that satisfy the inequality are greater than or equal to $-\frac{3}{5}$. In Case 2, the values of $x$ that satisfy the inequality are less than or equal to $-2$.

Q: How do I simplify the right-hand side of the inequality in Case 1?

A: To simplify the right-hand side of the inequality in Case 1, you need to subtract $\frac{13}{10}$ from both sides of the inequality.

Q: How do I simplify the right-hand side of the inequality in Case 2?

A: To simplify the right-hand side of the inequality in Case 2, you need to subtract $\frac{13}{10}$ from both sides of the inequality.

Q: What is the final answer to the inequality $5x^2 + 13x \geq -6$?

A: The final answer to the inequality is:

• $x \leq -2 \text{ or } x \geq -\frac{3}{5}$

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the inequality $5x^2 + 13x \geq -6$. We hope that this guide has been helpful in clarifying any doubts you may have had about the solution.

Final Answer

The final answer to the inequality $5x^2 + 13x \geq -6$ is:

• $x \leq -2 \text{ or } x \geq -\frac{3}{5}$