True Or False: ( − 5 , − 2 (-5,-2 ( − 5 , − 2 ] Is A Solution To The Following System Of Inequalities.${ \begin{array}{l} y \ \textless \ -2x - 6 \ y \ \textgreater \ 5x + 18 \end{array} }$A. True B. False

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Introduction

In mathematics, solving systems of inequalities is a crucial concept that involves finding the intersection of multiple inequalities. These inequalities can be linear or non-linear, and they can be represented graphically on a coordinate plane. In this article, we will explore whether the point $(-5,-2)$ is a solution to the following system of inequalities:

$$\begin{array}{l} y < -2x - 6 \\ y > 5x + 18 \end{array}$$

Understanding the System of Inequalities

To determine whether the point $(-5,-2)$ is a solution to the system of inequalities, we need to understand the concept of inequalities and how they are represented graphically. Inequalities are mathematical statements that compare two expressions using greater than, less than, or equal to symbols. In this case, we have two inequalities:

$$y < -2x - 6$$

$$y > 5x + 18$$

Graphing the Inequalities

To visualize the system of inequalities, we can graph the two inequalities on a coordinate plane. The first inequality, $y < -2x - 6$, can be graphed by drawing a line with a slope of $-2$ and a y-intercept of $-6$. The region below this line represents the solution to the inequality.

The second inequality, $y > 5x + 18$, can be graphed by drawing a line with a slope of $5$ and a y-intercept of $18$. The region above this line represents the solution to the inequality.

Finding the Intersection of the Inequalities

To find the solution to the system of inequalities, we need to find the intersection of the two regions. This can be done by finding the point where the two lines intersect. To find the intersection point, we can set the two equations equal to each other and solve for $x$.

$$-2x - 6 = 5x + 18$$

Solving for $x$, we get:

$$-7x = 24$$

$$x = -\frac{24}{7}$$

Substituting this value of $x$ into one of the original equations, we can find the corresponding value of $y$.

$$y = -2\left(-\frac{24}{7}\right) - 6$$

$$y = \frac{48}{7} - 6$$

$$y = \frac{48}{7} - \frac{42}{7}$$

$$y = \frac{6}{7}$$

Determining Whether the Point is a Solution

Now that we have found the intersection point, we can determine whether the point $(-5,-2)$ is a solution to the system of inequalities. To do this, we need to check whether the point lies within the intersection region.

Since the point $(-5,-2)$ lies below the line $y = -2x - 6$ and above the line $y = 5x + 18$, it does not lie within the intersection region. Therefore, the point $(-5,-2)$ is not a solution to the system of inequalities.

Conclusion

In conclusion, the point $(-5,-2)$ is not a solution to the system of inequalities. This is because the point lies outside the intersection region of the two inequalities. Therefore, the correct answer is B. False.

Final Thoughts

Solving systems of inequalities is a crucial concept in mathematics that involves finding the intersection of multiple inequalities. By graphing the inequalities and finding the intersection point, we can determine whether a given point is a solution to the system. In this article, we have explored whether the point $(-5,-2)$ is a solution to the following system of inequalities:

$$\begin{array}{l} y < -2x - 6 \\ y > 5x + 18 \end{array}$$

We have found that the point $(-5,-2)$ is not a solution to the system of inequalities, and therefore, the correct answer is B. False.

References

• [1] "Systems of Inequalities" by Math Open Reference
• [2] "Graphing Inequalities" by Khan Academy
• [3] "Solving Systems of Inequalities" by Purplemath

Related Articles

• [1] "Solving Systems of Linear Equations"
• [2] "Graphing Linear Equations"
• [3] "Solving Quadratic Equations"

Tags

• Systems of Inequalities
• Graphing Inequalities
• Solving Systems of Inequalities
• Math
• Mathematics
• Algebra
• Geometry
• Coordinate Plane
• Inequalities
• Linear Equations
• Quadratic Equations
  

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Introduction

In our previous article, we explored whether the point $(-5,-2)$ is a solution to the following system of inequalities:

$$\begin{array}{l} y < -2x - 6 \\ y > 5x + 18 \end{array}$$

We found that the point $(-5,-2)$ is not a solution to the system of inequalities. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is a system of inequalities?

A system of inequalities is a set of two or more inequalities that are combined to form a single system. In this case, we have two inequalities:

$$y < -2x - 6$$

$$y > 5x + 18$$

Q2: How do I graph a system of inequalities?

To graph a system of inequalities, you need to graph each inequality separately and then find the intersection of the two regions. You can use a coordinate plane to graph the inequalities.

Q3: What is the intersection of two inequalities?

The intersection of two inequalities is the region where the two inequalities overlap. In this case, we found that the intersection point is $x = -\frac{24}{7}$ and $y = \frac{6}{7}$.

Q4: How do I determine whether a point is a solution to a system of inequalities?

To determine whether a point is a solution to a system of inequalities, you need to check whether the point lies within the intersection region. If the point lies within the intersection region, then it is a solution to the system of inequalities.

Q5: What is the difference between a system of linear equations and a system of inequalities?

A system of linear equations is a set of two or more linear equations that are combined to form a single system. A system of inequalities is a set of two or more inequalities that are combined to form a single system. In a system of linear equations, the goal is to find the point of intersection, while in a system of inequalities, the goal is to find the region of intersection.

Q6: How do I solve a system of inequalities?

To solve a system of inequalities, you need to graph each inequality separately and then find the intersection of the two regions. You can use a coordinate plane to graph the inequalities.

Q7: What is the importance of solving systems of inequalities?

Solving systems of inequalities is an important concept in mathematics that has many real-world applications. It is used in fields such as economics, finance, and engineering to model and solve complex problems.

Q8: Can I use a calculator to solve a system of inequalities?

Yes, you can use a calculator to solve a system of inequalities. However, it is often more effective to graph the inequalities and find the intersection region by hand.

Conclusion

In conclusion, solving systems of inequalities is a crucial concept in mathematics that involves finding the intersection of multiple inequalities. By graphing the inequalities and finding the intersection point, we can determine whether a given point is a solution to the system. In this article, we have answered some frequently asked questions related to the topic.

Final Thoughts

Solving systems of inequalities is a complex concept that requires a deep understanding of mathematics. By practicing and applying the concepts, you can become proficient in solving systems of inequalities and apply them to real-world problems.

References

• [1] "Systems of Inequalities" by Math Open Reference
• [2] "Graphing Inequalities" by Khan Academy
• [3] "Solving Systems of Inequalities" by Purplemath

Related Articles

• [1] "Solving Systems of Linear Equations"
• [2] "Graphing Linear Equations"
• [3] "Solving Quadratic Equations"

Tags

• Systems of Inequalities
• Graphing Inequalities
• Solving Systems of Inequalities
• Math
• Mathematics
• Algebra
• Geometry
• Coordinate Plane
• Inequalities
• Linear Equations
• Quadratic Equations