Graph The Solution To This Inequality On The Number Line:$\[ 2x - 6 \ \textgreater \ -16 \\] $\[ 3x - 10 \leq 8 \\] Use The Drawing Tools To Form The Correct Answer On The Number Line.

Mar 1, 2025 by ADMIN 189 views

Graph the Solution to the Inequality on the Number Line

Introduction to Inequalities

Inequalities are mathematical expressions that compare two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. In this article, we will focus on graphing the solution to a system of linear inequalities on the number line. We will use the given inequalities: $2x - 6 > -16$ and $3x - 10 \leq 8$ .

Understanding the First Inequality

The first inequality is $2x - 6 > -16$ . To solve this inequality, we need to isolate the variable $x$ . We can do this by adding $6$ to both sides of the inequality, which gives us $2x > -10$ . Next, we divide both sides of the inequality by $2$ , resulting in $x > -5$ .

Understanding the Second Inequality

The second inequality is $3x - 10 \leq 8$ . To solve this inequality, we need to isolate the variable $x$ . We can do this by adding $10$ to both sides of the inequality, which gives us $3x \leq 18$ . Next, we divide both sides of the inequality by $3$ , resulting in $x \leq 6$ .

Graphing the Solution on the Number Line

Now that we have solved both inequalities, we can graph the solution on the number line. The first inequality, $x > -5$ , indicates that the solution is all real numbers greater than $-5$ . This can be represented on the number line as an open circle at $-5$ and an arrow pointing to the right.

The second inequality, $x \leq 6$ , indicates that the solution is all real numbers less than or equal to $6$ . This can be represented on the number line as a closed circle at $6$ and an arrow pointing to the left.

Finding the Intersection of the Two Inequalities

To find the intersection of the two inequalities, we need to find the values of $x$ that satisfy both inequalities. We can do this by finding the overlap between the two solution sets.

The first inequality, $x > -5$ , indicates that the solution is all real numbers greater than $-5$ . The second inequality, $x \leq 6$ , indicates that the solution is all real numbers less than or equal to $6$ .

Conclusion

In conclusion, we have graphed the solution to the system of linear inequalities on the number line. We have found that the intersection of the two inequalities is the set of all real numbers greater than $-5$ and less than or equal to $6$ . This can be represented on the number line as an open circle at $-5$ and a closed circle at $6$ , with an arrow pointing to the right.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Solve the first inequality: $2x - 6 > -16$ $2 x - 6 > - 16$
- Add $6$ to both sides: $2x > -10$
- Divide both sides by $2$ : $x > -5$
Solve the second inequality: $3x - 10 \leq 8$ $3 x - 10 \leq 8$
- Add $10$ to both sides: $3x \leq 18$
- Divide both sides by $3$ : $x \leq 6$
Graph the solution on the number line:
- Draw an open circle at $-5$ and an arrow pointing to the right
- Draw a closed circle at $6$ and an arrow pointing to the left
Find the intersection of the two inequalities:
- Find the overlap between the two solution sets
- The intersection is the set of all real numbers greater than $-5$ and less than or equal to $6$

Frequently Asked Questions

Q: What is the solution to the first inequality? A: The solution to the first inequality is $x > -5$ .
Q: What is the solution to the second inequality? A: The solution to the second inequality is $x \leq 6$ .
Q: What is the intersection of the two inequalities? A: The intersection of the two inequalities is the set of all real numbers greater than $-5$ and less than or equal to $6$ .

Final Answer

The final answer is $\boxed{(-5, 6]}$ .

Introduction

In our previous article, we graphed the solution to a system of linear inequalities on the number line. We solved the inequalities $2x - 6 > -16$ and $3x - 10 \leq 8$ and found the intersection of the two inequalities. In this article, we will answer some frequently asked questions about graphing the solution to inequalities on the number line.

Q&A

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

A: An open circle on the number line represents a value that is not included in the solution set, while a closed circle represents a value that is included in the solution set.

Q: How do I determine whether to use an open circle or a closed circle on the number line?

A: To determine whether to use an open circle or a closed circle, look at the inequality symbol. If the symbol is greater than (>), less than (<), or greater than or equal to (≥), use an open circle. If the symbol is less than or equal to (≤), use a closed circle.

Q: What is the purpose of the arrow on the number line?

A: The arrow on the number line indicates the direction of the solution set. If the arrow points to the right, the solution set includes all values greater than the point on the number line. If the arrow points to the left, the solution set includes all values less than the point on the number line.

Q: How do I find the intersection of two inequalities on the number line?

A: To find the intersection of two inequalities on the number line, look for the overlap between the two solution sets. The intersection is the set of values that satisfy both inequalities.

Q: What if the two inequalities have no intersection?

A: If the two inequalities have no intersection, it means that the solution sets do not overlap. In this case, the intersection is the empty set, represented by a pair of parentheses: ( ).

Q: Can I use the number line to solve systems of linear equations?

A: Yes, you can use the number line to solve systems of linear equations. However, you will need to use a different method to find the intersection of the two equations.

Q: How do I graph the solution to a system of linear inequalities with multiple variables?

A: To graph the solution to a system of linear inequalities with multiple variables, you will need to use a three-dimensional number line or a graphing calculator.