Solution Set Of Compound Inequality X Less Than 6 Or X Greater Than 10

This article aims to provide a comprehensive understanding of compound inequalities, solution sets, and how to determine which sets of numbers satisfy a given compound inequality. Specifically, we will address the question: Which set of numbers is included as part of the solution set of the compound inequality $x<6$ or $x>10$? We will explore the underlying concepts, break down the problem step-by-step, and arrive at the correct solution. This guide is designed to be accessible to anyone, regardless of their mathematical background, and will serve as a valuable resource for mastering compound inequalities.

Defining Compound Inequalities

To effectively tackle the given problem, it's crucial to first understand what compound inequalities are. Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." These connecting words play a critical role in determining the solution set.

Let's delve deeper into the two types of compound inequalities:

• "And" Compound Inequalities: These inequalities require that both conditions be true simultaneously. The solution set consists of all values that satisfy both inequalities. Graphically, this is represented by the intersection of the solution sets of the individual inequalities. For example, the compound inequality "$x > 2$ and $x < 5$" means that $x$ must be greater than 2 and less than 5. The solution set would include all numbers between 2 and 5, but not including 2 and 5 themselves.

• "Or" Compound Inequalities: These inequalities require that at least one of the conditions be true. The solution set consists of all values that satisfy either inequality or both. Graphically, this is represented by the union of the solution sets of the individual inequalities. For instance, the compound inequality "$x < 1$ or $x > 4$" means that $x$ must be less than 1 or greater than 4. The solution set would include all numbers less than 1 and all numbers greater than 4.

The key difference lies in the connecting word. "And" implies an intersection, while "or" implies a union. Understanding this distinction is paramount when solving compound inequalities.

To solidify your understanding, consider these additional examples:

• Example 1: "And" Inequality

  • Inequality: $-3 extless x extless 2$ (This can be rewritten as $x > -3$ and $x < 2$)
  • Solution: All numbers greater than -3 and less than 2. On a number line, this would be the segment between -3 and 2, excluding the endpoints.

• Example 2: "Or" Inequality

  • Inequality: $x extless -1$ or $x extgreater 3$
  • Solution: All numbers less than -1 or greater than 3. On a number line, this would be two separate rays extending to the left from -1 and to the right from 3.

Visualizing these inequalities on a number line can be incredibly helpful. Draw a number line and shade the regions that satisfy each inequality. For "and" inequalities, the overlapping shaded region represents the solution. For "or" inequalities, the combined shaded regions represent the solution.

By grasping the fundamental concepts of compound inequalities and the crucial role of "and" and "or," you'll be well-equipped to tackle a wide range of problems, including the one presented in this article.

Understanding Solution Sets

Before we directly address the problem, let's solidify our understanding of solution sets. A solution set, in the context of inequalities (or equations), is the set of all values that make the inequality (or equation) true. In simpler terms, it's the collection of numbers that, when substituted for the variable, satisfy the given condition.

For a single inequality, the solution set might be an interval of numbers, a single number, or even the empty set (meaning no solutions exist). For example:

• Inequality: $x > 5$

  • Solution Set: All numbers greater than 5. This can be represented in interval notation as $(5, ext{ ∞})$.

• Inequality: $x ext{≤} -2$

  • Solution Set: All numbers less than or equal to -2. This can be represented in interval notation as $(- ext{∞}, -2]$.

• Equation: $x + 3 = 7$

  • Solution Set: The single number 4, as it's the only value that makes the equation true.

• Inequality: $x^2 < 0$

  • Solution Set: Empty set, as there is no real number whose square is negative.

When dealing with compound inequalities, the solution set is determined by the connecting word, as we discussed earlier.

• "And" Compound Inequality: The solution set is the intersection of the solution sets of the individual inequalities. This means a number must satisfy both inequalities to be part of the solution set.

• "Or" Compound Inequality: The solution set is the union of the solution sets of the individual inequalities. This means a number must satisfy at least one of the inequalities to be part of the solution set.

Consider these examples to illustrate the concept of solution sets for compound inequalities:

• Example 1: "And" Inequality

  • Inequality: $1 ext{≤} x ext{≤} 4$ (equivalent to $x ext{≥} 1$ and $x ext{≤} 4$)
  • Solution Set: All numbers between 1 and 4, inclusive. In interval notation, this is represented as $[1, 4]$.

• Example 2: "Or" Inequality

  • Inequality: $x < -2$ or $x > 2$
  • Solution Set: All numbers less than -2 or greater than 2. In interval notation, this is represented as $(- ext{∞}, -2) ext{∪} (2, ext{ ∞})$.

Understanding solution sets is crucial for determining whether a given set of numbers is "included as part of" the solution set of an inequality. This involves checking if each number in the given set satisfies the inequality (or at least one of the inequalities in the case of an "or" compound inequality).

In the context of our main problem, we will be given sets of numbers and need to determine which set (or sets) contains only numbers that satisfy the compound inequality $x < 6$ or $x > 10$. This requires us to apply our understanding of both compound inequalities and solution sets.

Solving the Compound Inequality $x<6$ or $x>10$

Now, let's focus on the specific compound inequality presented in the problem: $x < 6$ or $x > 10$. This is an "or" compound inequality, meaning a number is part of the solution set if it satisfies either $x < 6$ or $x > 10$ (or both).

Let's break down the individual inequalities:

• $x < 6$: This inequality represents all numbers that are strictly less than 6. On a number line, this would be an open interval extending to the left from 6.

• $x > 10$: This inequality represents all numbers that are strictly greater than 10. On a number line, this would be an open interval extending to the right from 10.

Since this is an "or" inequality, the solution set is the union of the solution sets of these two individual inequalities. This means any number that is either less than 6 or greater than 10 is part of the solution set.

It's helpful to visualize this on a number line. Imagine a number line with an open circle at 6 and an arrow extending to the left, representing $x < 6$. Then, imagine another open circle at 10 and an arrow extending to the right, representing $x > 10$. The entire region covered by these two arrows represents the solution set of the compound inequality.

In interval notation, the solution set can be expressed as $(- ext{∞}, 6) ext{∪} (10, ext{ ∞})$. This notation clearly shows that the solution set consists of two separate intervals: all numbers from negative infinity up to (but not including) 6, and all numbers from 10 (not including 10) up to positive infinity.

Numbers between 6 and 10 (inclusive) are not part of the solution set. This is a crucial point to remember when checking the given sets of numbers.

To further solidify your understanding, consider some examples:

• Is 5 part of the solution set? Yes, because 5 < 6.
• Is 12 part of the solution set? Yes, because 12 > 10.
• Is 8 part of the solution set? No, because 8 is not less than 6 and 8 is not greater than 10.
• Is 6 part of the solution set? No, because the inequality is strictly less than 6 ($x < 6$), so 6 itself is not included.
• Is 10 part of the solution set? No, because the inequality is strictly greater than 10 ($x > 10$), so 10 itself is not included.

With a clear understanding of the solution set for $x < 6$ or $x > 10$, we are now ready to examine the given sets of numbers and determine which one is entirely included within this solution set.

Analyzing the Given Sets of Numbers

Now comes the crucial step of analyzing the given sets of numbers to determine which one is entirely included in the solution set of the compound inequality $x < 6$ or $x > 10$. Remember, a set is included in the solution set if every number in the set satisfies the compound inequality.

Let's examine each set individually:

• Set 1: $\{-7, -1.7, 6.1, 10\}$

  • -7: Satisfies $x < 6$ (since -7 < 6)
  • -1.7: Satisfies $x < 6$ (since -1.7 < 6)
  • 6.1: Does not satisfy $x < 6$ (since 6.1 > 6) and does not satisfy $x > 10$ (since 6.1 < 10). This number disqualifies the entire set.
  • 10: Does not satisfy $x < 6$ (since 10 > 6) and does not satisfy $x > 10$ (since 10 is not strictly greater than 10). This number further confirms the set is not a solution.

• Set 2: $\{-3, 4.5, 13.6, 19\}$

  • -3: Satisfies $x < 6$ (since -3 < 6)
  • 4.5: Satisfies $x < 6$ (since 4.5 < 6)
  • 13.6: Satisfies $x > 10$ (since 13.6 > 10)
  • 19: Satisfies $x > 10$ (since 19 > 10)
  • All numbers in this set satisfy the compound inequality.

• Set 3: $\{0, 6, 9.8, 14\}$

  • 0: Satisfies $x < 6$ (since 0 < 6)
  • 6: Does not satisfy $x < 6$ (since 6 is not strictly less than 6) and does not satisfy $x > 10$ (since 6 < 10). This number disqualifies the entire set.
  • 9.8: Does not satisfy $x < 6$ (since 9.8 > 6) and does not satisfy $x > 10$ (since 9.8 < 10). This number further confirms the set is not a solution.
  • 14: Satisfies $x > 10$ (since 14 > 10)

• Set 4: $\{8.5, 9.1\}$

  • 8. 5: Does not satisfy $x < 6$ (since 8.5 > 6) and does not satisfy $x > 10$ (since 8.5 < 10).
  • 9. 1: Does not satisfy $x < 6$ (since 9.1 > 6) and does not satisfy $x > 10$ (since 9.1 < 10).
  • Neither number in this set satisfies the compound inequality.

Conclusion: Identifying the Solution Set

After carefully analyzing each set of numbers, we have determined that only one set is entirely included in the solution set of the compound inequality $x < 6$ or $x > 10$.

The correct answer is Set 2: $\{-3, 4.5, 13.6, 19\}$.

All numbers in this set (-3, 4.5, 13.6, and 19) satisfy the compound inequality. The numbers -3 and 4.5 are less than 6, while the numbers 13.6 and 19 are greater than 10.

The other sets were disqualified because they contained at least one number that did not satisfy the compound inequality. This highlights the importance of checking every number in a set when determining if it's part of the solution set of an inequality.

This problem demonstrates the importance of understanding compound inequalities, solution sets, and the logical connectives "and" and "or." By breaking down the problem into smaller steps and systematically analyzing each option, we were able to arrive at the correct solution.

Remember, practice is key to mastering mathematical concepts. Try working through similar problems to further solidify your understanding of compound inequalities and solution sets. You can also explore more complex inequalities and systems of inequalities to challenge yourself and expand your mathematical skills.

This comprehensive guide should provide you with a solid foundation for understanding and solving problems involving compound inequalities. Keep practicing, and you'll be well-equipped to tackle any inequality challenge!