Solve $-\frac{2}{3} N \leq 16$. Which Of The Following Must Be True About The Inequality And The Resulting Graph? Select Three Options.A. $n \leq -24$ B. $n \geq -24$ C. The Circle Is Open. D. The Circle Is Closed. E. The

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. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will solve the inequality $-\frac{2}{3} n \leq 16$ and discuss the resulting graph.

Step 1: Multiply Both Sides by -1/3

To solve the inequality, we need to isolate the variable $n$. The first step is to multiply both sides of the inequality by $-\frac{2}{3}$. This will eliminate the fraction and make it easier to work with.

-\frac{2}{3} n \leq 16
\Rightarrow n \geq -24

Step 2: Simplify the Inequality

After multiplying both sides by $-\frac{2}{3}$, we get $n \geq -24$. This is the simplified form of the inequality.

Step 3: Graph the Inequality

To graph the inequality, we need to draw a number line and mark the point $-24$ with an open circle. This is because the inequality is greater than or equal to, and the point $-24$ is included in the solution set.

n \geq -24

Graph:

The graph of the inequality is a closed circle at $-24$ and all points to the right of $-24$.

Discussion

Now that we have solved the inequality and graphed it, let's discuss the options.

Option A: $n \leq -24$

This option is incorrect because the inequality is greater than or equal to, not less than or equal to.

Option B: $n \geq -24$

This option is correct because the inequality is greater than or equal to, and the point $-24$ is included in the solution set.

Option C: The circle is open.

This option is incorrect because the circle is closed, not open.

Option D: The circle is closed.

This option is correct because the circle is closed, and the point $-24$ is included in the solution set.

Option E: The graph is a line.

This option is incorrect because the graph is a closed circle at $-24$ and all points to the right of $-24$.

Conclusion

In conclusion, the correct options are B and D. The inequality $-\frac{2}{3} n \leq 16$ is solved by multiplying both sides by $-\frac{2}{3}$, resulting in $n \geq -24$. The graph of the inequality is a closed circle at $-24$ and all points to the right of $-24$.

Final Answer

The final answer is:

• B: $n \geq -24$
• D: The circle is closed.
  Solving Inequalities: A Q&A Guide =====================================

Introduction

In our previous article, we solved the inequality $-\frac{2}{3} n \leq 16$ and discussed the resulting graph. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality sign. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: How do I know if I should multiply or divide both sides of the inequality by a negative number?

A: When multiplying or dividing both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have $n \leq 5$ and you multiply both sides by $-2$, you get $-2n \geq -10$.

Q: What is the difference between a closed circle and an open circle on a graph?

A: A closed circle on a graph represents a point that is included in the solution set, while an open circle represents a point that is not included in the solution set.

Q: How do I determine if a point is included in the solution set or not?

A: To determine if a point is included in the solution set or not, you need to look at the inequality sign. If the inequality sign is greater than or equal to (≥), the point is included in the solution set. If the inequality sign is less than or equal to (≤), the point is included in the solution set. If the inequality sign is greater than (>) or less than (<), the point is not included in the solution set.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \leq 0$ or $\frac{f(x)}{g(x)} \geq 0$, where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) \leq 0$ or $f(x) \geq 0$, where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the zeros of the numerator and denominator and then use the sign of the rational expression to determine the solution set.

Conclusion

In conclusion, solving inequalities involves isolating the variable on one side of the inequality sign and then using the sign of the inequality to determine the solution set. By following these steps, you can solve a wide range of inequalities, from linear inequalities to rational inequalities.

Final Answer

The final answer is:

• The first step in solving an inequality is to isolate the variable on one side of the inequality sign.
• When multiplying or dividing both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.
• A closed circle on a graph represents a point that is included in the solution set, while an open circle represents a point that is not included in the solution set.
• To determine if a point is included in the solution set or not, you need to look at the inequality sign.
• A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants.
• To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.
• A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \leq 0$ or $\frac{f(x)}{g(x)} \geq 0$, where $f(x)$ and $g(x)$ are polynomials.
• To solve a rational inequality, you need to find the zeros of the numerator and denominator and then use the sign of the rational expression to determine the solution set.