Given That The Solutions To $x^2-x-26=-6$ Are $x=5$ And \$x=-4$[/tex\], Which Is A Reasonable Solution To The Inequality $x^2-x-26\ \textgreater \ -6$?A. $-4\ \textless \ X\ \textless \ 5$ B.

Introduction

When solving inequalities, it's essential to understand the relationship between the solutions of the corresponding equation and the inequality. In this article, we will explore how to find a reasonable solution to the inequality $x^2-x-26 > -6$, given that the solutions to the equation $x^2-x-26=-6$ are $x=5$ and $x=-4$.

Understanding the Relationship Between Equations and Inequalities

To solve the inequality $x^2-x-26 > -6$, we need to understand the relationship between the solutions of the corresponding equation $x^2-x-26=-6$ and the inequality. The equation $x^2-x-26=-6$ has two solutions, $x=5$ and $x=-4$. These solutions are the points where the graph of the equation intersects the x-axis.

Graphing the Equation

To visualize the relationship between the equation and the inequality, let's graph the equation $x^2-x-26=-6$. We can rewrite the equation as $x^2-x-20=0$. This is a quadratic equation that can be factored as $(x-5)(x+4)=0$. The graph of the equation is a parabola that intersects the x-axis at $x=5$ and $x=-4$.

Understanding the Inequality

The inequality $x^2-x-26 > -6$ is a quadratic inequality that can be rewritten as $x^2-x-20 > 0$. This inequality is true for values of $x$ that make the quadratic expression positive. To find the values of $x$ that satisfy the inequality, we need to find the intervals where the quadratic expression is positive.

Finding the Intervals

To find the intervals where the quadratic expression $x^2-x-20$ is positive, we need to find the roots of the equation $x^2-x-20=0$. The roots of the equation are $x=5$ and $x=-4$. These roots divide the number line into three intervals: $(-\infty, -4)$, $(-4, 5)$, and $(5, \infty)$.

Testing the Intervals

To determine which intervals satisfy the inequality $x^2-x-20 > 0$, we need to test each interval by substituting a test value into the quadratic expression. Let's test each interval:

• For the interval $(-\infty, -4)$, let's choose a test value $x=-5$. Substituting $x=-5$ into the quadratic expression, we get $(-5)^2-(-5)-20=25+5-20=10$. Since $10>0$, the interval $(-\infty, -4)$ satisfies the inequality.
• For the interval $(-4, 5)$, let's choose a test value $x=0$. Substituting $x=0$ into the quadratic expression, we get $(0)^2-(0)-20=-20$. Since $-20<0$, the interval $(-4, 5)$ does not satisfy the inequality.
• For the interval $(5, \infty)$, let's choose a test value $x=6$. Substituting $x=6$ into the quadratic expression, we get $(6)^2-(6)-20=36-6-20=10$. Since $10>0$, the interval $(5, \infty)$ satisfies the inequality.

Conclusion

Based on the analysis, we can conclude that the intervals $(-\infty, -4)$ and $(5, \infty)$ satisfy the inequality $x^2-x-26 > -6$. Therefore, a reasonable solution to the inequality is $x \in (-\infty, -4) \cup (5, \infty)$.

Final Answer

The final answer is: $\boxed{(-\infty, -4) \cup (5, \infty)}$

Introduction

When solving inequalities, it's essential to understand the relationship between the solutions of the corresponding equation and the inequality. In this article, we will explore how to find a reasonable solution to the inequality $x^2-x-26 > -6$, given that the solutions to the equation $x^2-x-26=-6$ are $x=5$ and $x=-4$.

Understanding the Relationship Between Equations and Inequalities

To solve the inequality $x^2-x-26 > -6$, we need to understand the relationship between the solutions of the corresponding equation $x^2-x-26=-6$ and the inequality. The equation $x^2-x-26=-6$ has two solutions, $x=5$ and $x=-4$. These solutions are the points where the graph of the equation intersects the x-axis.

Graphing the Equation

To visualize the relationship between the equation and the inequality, let's graph the equation $x^2-x-26=-6$. We can rewrite the equation as $x^2-x-20=0$. This is a quadratic equation that can be factored as $(x-5)(x+4)=0$. The graph of the equation is a parabola that intersects the x-axis at $x=5$ and $x=-4$.

Understanding the Inequality

The inequality $x^2-x-26 > -6$ is a quadratic inequality that can be rewritten as $x^2-x-20 > 0$. This inequality is true for values of $x$ that make the quadratic expression positive. To find the values of $x$ that satisfy the inequality, we need to find the intervals where the quadratic expression is positive.

Finding the Intervals

To find the intervals where the quadratic expression $x^2-x-20$ is positive, we need to find the roots of the equation $x^2-x-20=0$. The roots of the equation are $x=5$ and $x=-4$. These roots divide the number line into three intervals: $(-\infty, -4)$, $(-4, 5)$, and $(5, \infty)$.

Testing the Intervals

To determine which intervals satisfy the inequality $x^2-x-20 > 0$, we need to test each interval by substituting a test value into the quadratic expression. Let's test each interval:

• For the interval $(-\infty, -4)$, let's choose a test value $x=-5$. Substituting $x=-5$ into the quadratic expression, we get $(-5)^2-(-5)-20=25+5-20=10$. Since $10>0$, the interval $(-\infty, -4)$ satisfies the inequality.
• For the interval $(-4, 5)$, let's choose a test value $x=0$. Substituting $x=0$ into the quadratic expression, we get $(0)^2-(0)-20=-20$. Since $-20<0$, the interval $(-4, 5)$ does not satisfy the inequality.
• For the interval $(5, \infty)$, let's choose a test value $x=6$. Substituting $x=6$ into the quadratic expression, we get $(6)^2-(6)-20=36-6-20=10$. Since $10>0$, the interval $(5, \infty)$ satisfies the inequality.

Conclusion

Based on the analysis, we can conclude that the intervals $(-\infty, -4)$ and $(5, \infty)$ satisfy the inequality $x^2-x-26 > -6$. Therefore, a reasonable solution to the inequality is $x \in (-\infty, -4) \cup (5, \infty)$.

Q&A

Q: What is the relationship between the solutions of the equation and the inequality?

A: The solutions of the equation are the points where the graph of the equation intersects the x-axis. The inequality is true for values of $x$ that make the quadratic expression positive.

Q: How do we find the intervals where the quadratic expression is positive?

A: We need to find the roots of the equation $x^2-x-20=0$. The roots of the equation are $x=5$ and $x=-4$. These roots divide the number line into three intervals: $(-\infty, -4)$, $(-4, 5)$, and $(5, \infty)$.

Q: How do we test each interval to determine which intervals satisfy the inequality?

A: We need to substitute a test value into the quadratic expression for each interval. If the result is positive, the interval satisfies the inequality.

Q: What are the intervals that satisfy the inequality $x^2-x-26 > -6$?

A: The intervals $(-\infty, -4)$ and $(5, \infty)$ satisfy the inequality.

Q: What is a reasonable solution to the inequality $x^2-x-26 > -6$?

A: A reasonable solution to the inequality is $x \in (-\infty, -4) \cup (5, \infty)$.

Final Answer

The final answer is: $\boxed{(-\infty, -4) \cup (5, \infty)}$