Solve And Write Interval Notation For The Solution Set. Then Graph The Solution Set.$\[ 2x \leq -16 \text{ Or } X - 8 \ \textgreater \ 0 \\]Select The Correct Choice Below And Fill In Any Answer Boxes In Your Choice.A. The Solution Set Is

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Introduction

In mathematics, inequalities are used to describe relationships between variables. Solving and graphing inequalities in interval notation is an essential skill in algebra and mathematics. In this article, we will learn how to solve and write interval notation for the solution set, and then graph the solution set.

Understanding Interval Notation

Interval notation is a way of writing the solution set of an inequality. It consists of a set of numbers that satisfy the inequality. The solution set can be written in the form of an interval, which is a range of values that include all the numbers that satisfy the inequality.

Solving the Inequality

The given inequality is:

2x≤−16 or x−8 \textgreater 0{ 2x \leq -16 \text{ or } x - 8 \ \textgreater \ 0 }

To solve this inequality, we need to solve each part separately.

Solving the First Part

The first part of the inequality is:

2x≤−16{ 2x \leq -16 }

To solve this inequality, we need to isolate the variable x. We can do this by dividing both sides of the inequality by 2.

2x <= -16
x <= -16/2
x <= -8

So, the solution set for the first part is x ≤ -8.

Solving the Second Part

The second part of the inequality is:

x−8 \textgreater 0{ x - 8 \ \textgreater \ 0 }

To solve this inequality, we need to isolate the variable x. We can do this by adding 8 to both sides of the inequality.

x - 8 > 0
x > 8

So, the solution set for the second part is x > 8.

Writing Interval Notation

Now that we have solved the inequality, we can write the solution set in interval notation. The solution set is the union of the two solution sets, which is x ≤ -8 or x > 8.

In interval notation, this can be written as:

(−∞,−8]∪(8,∞){ (-\infty, -8] \cup (8, \infty) }

Graphing the Solution Set

To graph the solution set, we need to graph the two solution sets separately and then combine them.

Graphing the First Part

The first part of the solution set is x ≤ -8. This can be graphed as a closed circle at x = -8 and a vertical line at x = -8.

Graphing the Second Part

The second part of the solution set is x > 8. This can be graphed as an open circle at x = 8 and a vertical line at x = 8.

Combining the Two Solution Sets

To combine the two solution sets, we need to graph the union of the two solution sets. This can be done by graphing the two solution sets separately and then combining them.

The final graph of the solution set is a combination of the two solution sets, which is a closed circle at x = -8 and an open circle at x = 8, with a vertical line at x = -8 and x = 8.

Conclusion

In this article, we learned how to solve and write interval notation for the solution set, and then graph the solution set. We solved the inequality 2x ≤ -16 or x - 8 > 0, and then wrote the solution set in interval notation. We also graphed the solution set, which is a combination of two solution sets.

Key Takeaways

  • Solving and graphing inequalities in interval notation is an essential skill in algebra and mathematics.
  • Interval notation is a way of writing the solution set of an inequality.
  • To solve an inequality, we need to isolate the variable x.
  • To graph the solution set, we need to graph the two solution sets separately and then combine them.

Practice Problems

  1. Solve the inequality 3x + 2 ≤ 5 and write the solution set in interval notation.
  2. Graph the solution set of the inequality x - 2 > 4.
  3. Solve the inequality 2x - 3 ≥ 5 and write the solution set in interval notation.

Answer Key

  1. The solution set is (-∞, -1] ∪ (3, ∞).
  2. The graph of the solution set is an open circle at x = 6 and a vertical line at x = 6.
  3. The solution set is (3, ∞).
    Solving and Graphing Inequalities in Interval Notation: Q&A ===========================================================

Introduction

In our previous article, we learned how to solve and write interval notation for the solution set, and then graph the solution set. In this article, we will answer some frequently asked questions about solving and graphing inequalities in interval notation.

Q&A

Q: What is interval notation?

A: Interval notation is a way of writing the solution set of an inequality. It consists of a set of numbers that satisfy the inequality.

Q: How do I solve an inequality in interval notation?

A: To solve an inequality in interval notation, you need to isolate the variable x. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the difference between a closed circle and an open circle in graphing?

A: A closed circle 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 graph the solution set of an inequality?

A: To graph the solution set of an inequality, you need to graph the two solution sets separately and then combine them. You can use a closed circle to represent a point that is included in the solution set and an open circle to represent a point that is not included in the solution set.

Q: What is the union of two solution sets?

A: The union of two solution sets is the combination of the two solution sets. It includes all the numbers that satisfy either of the two solution sets.

Q: How do I write the solution set in interval notation?

A: To write the solution set in interval notation, you need to use the following symbols:

  • (-∞, a] represents all numbers less than or equal to a
  • (a, ∞) represents all numbers greater than a
  • (-∞, a) represents all numbers less than a
  • (a, ∞) represents all numbers greater than a

Q: What is the difference between a union and an intersection of two solution sets?

A: The union of two solution sets includes all the numbers that satisfy either of the two solution sets, while the intersection of two solution sets includes all the numbers that satisfy both of the two solution sets.

Q: How do I graph the intersection of two solution sets?

A: To graph the intersection of two solution sets, you need to graph the two solution sets separately and then find the common points between them.

Practice Problems

  1. Solve the inequality 2x + 3 ≤ 5 and write the solution set in interval notation.
  2. Graph the solution set of the inequality x - 4 > 2.
  3. Solve the inequality 3x - 2 ≥ 7 and write the solution set in interval notation.

Answer Key

  1. The solution set is (-∞, -1] ∪ (3, ∞).
  2. The graph of the solution set is an open circle at x = 6 and a vertical line at x = 6.
  3. The solution set is (3, ∞).

Conclusion

In this article, we answered some frequently asked questions about solving and graphing inequalities in interval notation. We covered topics such as interval notation, solving inequalities, graphing solution sets, and writing solution sets in interval notation.

Key Takeaways

  • Interval notation is a way of writing the solution set of an inequality.
  • To solve an inequality, you need to isolate the variable x.
  • To graph the solution set, you need to graph the two solution sets separately and then combine them.
  • The union of two solution sets includes all the numbers that satisfy either of the two solution sets.
  • The intersection of two solution sets includes all the numbers that satisfy both of the two solution sets.

Resources

  • Khan Academy: Solving Inequalities
  • Mathway: Solving Inequalities
  • Wolfram Alpha: Solving Inequalities

Final Thoughts

Solving and graphing inequalities in interval notation is an essential skill in algebra and mathematics. By understanding interval notation, solving inequalities, and graphing solution sets, you can solve a wide range of problems in mathematics and other fields.