Solving Compound Inequalities A Comprehensive Guide To 6x + 8 ≤ 20 Or 5 + 4x ≥ 33

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Introduction to Compound Inequalities

When it comes to solving inequalities, we often encounter scenarios where we need to deal with more than one inequality at the same time. These are known as compound inequalities. In this comprehensive guide, we will delve deep into the process of solving compound inequalities, focusing specifically on inequalities connected by the words "or" and "and." Our primary example will be the compound inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$. Understanding how to solve such inequalities is crucial for various mathematical applications and real-world problem-solving.

Compound inequalities involve combining two or more inequalities using logical connectives. The two most common connectives are "or" and "and." When inequalities are connected by "or," it means we are looking for solutions that satisfy either one inequality or the other, or both. On the other hand, when inequalities are connected by "and," we are looking for solutions that satisfy both inequalities simultaneously. This distinction is vital because the solution set will vary significantly depending on the connective used.

The inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$ is an example of a compound inequality connected by "or." To solve this, we will tackle each inequality separately and then combine their solution sets. This involves isolating the variable $x$ in each inequality, which will give us a clear range of values that satisfy each condition. Once we have the individual solution sets, we will merge them according to the "or" condition, meaning we will include any value that satisfies either inequality. This process will give us the final solution set, which we can represent graphically on a number line and express in interval notation.

The ability to solve compound inequalities is a fundamental skill in algebra and is essential for understanding more advanced mathematical concepts. By mastering this skill, you will be better equipped to tackle a wide range of problems, from basic algebraic equations to complex calculus problems. This guide will provide you with a step-by-step approach to solving compound inequalities, ensuring you grasp the underlying principles and can apply them confidently.

Solving $6x + 8 \leq 20$

To begin, let's focus on the first inequality: $6x + 8 \leq 20$. The primary goal here is to isolate the variable $x$ on one side of the inequality. This involves a series of algebraic manipulations that maintain the inequality's balance. The steps we take are similar to those used in solving equations, but with a crucial difference: we must be mindful of how certain operations affect the direction of the inequality sign, especially when multiplying or dividing by a negative number.

The first step in solving this inequality is to subtract 8 from both sides. This operation helps to isolate the term with $x$ on one side. Performing this subtraction, we get:

$6x + 8 - 8 \leq 20 - 8$

Which simplifies to:

$6x \leq 12$

Now that we have $6x \leq 12$, the next step is to divide both sides by 6 to isolate $x$. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged. Dividing both sides by 6, we get:

$\frac{6x}{6} \leq \frac{12}{6}$

This simplifies to:

$x \leq 2$

So, the solution to the first inequality, $6x + 8 \leq 20$, is $x \leq 2$. This means that any value of $x$ that is less than or equal to 2 will satisfy the inequality. We can represent this solution set graphically on a number line. To do this, we would draw a closed circle at 2 (indicating that 2 is included in the solution) and shade the region to the left of 2, representing all values less than 2.

In interval notation, the solution $x \leq 2$ is expressed as $(-\infty, 2]$. The parenthesis on the left side indicates that negative infinity is not included in the interval (as it is a concept, not a number), and the square bracket on the right side indicates that 2 is included in the interval. Understanding how to represent solutions in interval notation and graphically is essential for comprehending the full scope of the solution set and for communicating it effectively.

Solving $5 + 4x \geq 33$

Next, we turn our attention to the second inequality: $5 + 4x \geq 33$. Just as with the first inequality, our goal here is to isolate the variable $x$ on one side. This involves performing algebraic operations that maintain the integrity of the inequality while moving us closer to isolating $x$.

The first step in solving this inequality is to subtract 5 from both sides. This will help us isolate the term containing $x$. Performing this subtraction, we get:

$5 + 4x - 5 \geq 33 - 5$

Which simplifies to:

$4x \geq 28$

Now that we have $4x \geq 28$, the next step is to divide both sides by 4 to isolate $x$. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged. Dividing both sides by 4, we get:

$\frac{4x}{4} \geq \frac{28}{4}$

This simplifies to:

$x \geq 7$

Thus, the solution to the second inequality, $5 + 4x \geq 33$, is $x \geq 7$. This means that any value of $x$ that is greater than or equal to 7 will satisfy the inequality. To represent this solution graphically on a number line, we would draw a closed circle at 7 (indicating that 7 is included in the solution) and shade the region to the right of 7, representing all values greater than 7.

In interval notation, the solution $x \geq 7$ is expressed as $[7, \infty)$. The square bracket on the left side indicates that 7 is included in the interval, and the parenthesis on the right side indicates that positive infinity is not included in the interval. Understanding how to express solutions in interval notation and graphically provides a clear and concise way to communicate the solution set.

Combining the Solutions with "Or"

Now that we have solved each inequality individually, we need to combine their solutions to find the solution to the compound inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$. The key here is the word "or," which means we are looking for all values of $x$ that satisfy either the first inequality ($x \leq 2$) or the second inequality ($x \geq 7$), or both.

We found that the solution to $6x + 8 \leq 20$ is $x \leq 2$, and the solution to $5 + 4x \geq 33$ is $x \geq 7$. To combine these solutions, we consider all values that fall into either of these categories. This means we include all numbers less than or equal to 2, as well as all numbers greater than or equal to 7.

When combining solutions with "or," we essentially take the union of the individual solution sets. In this case, the union of the sets $x \leq 2$ and $x \geq 7$ includes all real numbers except those between 2 and 7. To visualize this, imagine two shaded regions on a number line: one shaded to the left of 2 (including 2) and another shaded to the right of 7 (including 7). There is a gap between these two shaded regions, representing the numbers that do not satisfy either inequality.

In interval notation, the solution to the compound inequality is expressed as $(-\infty, 2] \cup [7, \infty)$. The symbol $\cup$ represents the union of the two intervals, indicating that we are combining the elements of both sets. The interval $(-\infty, 2]$ includes all numbers less than or equal to 2, and the interval $[7, \infty)$ includes all numbers greater than or equal to 7. The union of these intervals gives us the complete solution set for the compound inequality.

Understanding how to combine solutions with "or" is crucial for solving compound inequalities. It requires a clear understanding of the union of sets and how to represent this union both graphically and in interval notation. This skill is essential for more advanced mathematical problem-solving and real-world applications.

Graphical Representation of the Solution

A graphical representation is an invaluable tool for understanding and communicating the solution to inequalities, especially compound inequalities. It provides a visual depiction of the solution set, making it easier to grasp the range of values that satisfy the inequality. For the compound inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$, a number line graph can clearly illustrate the solution.

To graph the solution on a number line, we first represent the individual solutions to each inequality. For $x \leq 2$, we draw a closed circle at 2, indicating that 2 is included in the solution, and shade the region to the left of 2. This shaded region represents all values less than or equal to 2. For $x \geq 7$, we draw a closed circle at 7, indicating that 7 is included in the solution, and shade the region to the right of 7. This shaded region represents all values greater than or equal to 7.

Since the compound inequality is connected by "or," we are looking for values that satisfy either $x \leq 2$ or $x \geq 7$. This means that the solution set includes all the shaded regions from both individual solutions. On the number line, this will appear as two separate shaded regions with a gap in between. The region between 2 and 7 is left unshaded, as these values do not satisfy either inequality.

The graph clearly shows the solution set: all numbers less than or equal to 2, and all numbers greater than or equal to 7. This visual representation can be particularly helpful in understanding the solution when dealing with more complex compound inequalities. It provides a quick and intuitive way to see which values are included in the solution and which are not.

Moreover, the graphical representation serves as a bridge to expressing the solution in interval notation. The two shaded regions on the number line correspond directly to the intervals $(-\infty, 2]$ and $[7, \infty)$. The gap between the regions highlights the fact that there are no solutions in the interval (2, 7). This visual aid can prevent errors when translating the solution into interval notation and ensures a clear understanding of the solution set.

Expressing the Solution in Interval Notation

Interval notation is a concise and widely used method for expressing the solution sets of inequalities. It provides a clear and unambiguous way to represent a range of values on the number line. For the compound inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$, expressing the solution in interval notation is a crucial step in fully understanding and communicating the result.

We have already determined that the solution to the compound inequality is $x \leq 2$ or $x \geq 7$. This means that the solution set consists of two separate intervals: one containing all numbers less than or equal to 2, and another containing all numbers greater than or equal to 7. To express these intervals in interval notation, we use parentheses and square brackets to indicate whether the endpoints are included in the solution set.

For the interval $x \leq 2$, we use a parenthesis on the left side to indicate that negative infinity is not included (as it is a concept, not a number), and a square bracket on the right side to indicate that 2 is included in the solution. This interval is written as $(-\infty, 2]$. The square bracket signifies that 2 is part of the solution set, while the parenthesis indicates that negative infinity is an unbounded concept and not a specific number.

For the interval $x \geq 7$, we use a square bracket on the left side to indicate that 7 is included in the solution, and a parenthesis on the right side to indicate that positive infinity is not included. This interval is written as $[7, \infty)$. The square bracket signifies that 7 is part of the solution set, while the parenthesis indicates that positive infinity is an unbounded concept.

Since the compound inequality is connected by "or," we need to combine these two intervals using the union symbol, $\cup$. The final solution in interval notation is $(-\infty, 2] \cup [7, \infty)$. This notation clearly communicates that the solution set includes all numbers less than or equal to 2, as well as all numbers greater than or equal to 7. There are no solutions in the interval between 2 and 7.

Understanding how to express solutions in interval notation is essential for advanced mathematical work. It provides a precise way to represent solution sets and is a standard convention in mathematical communication. Mastering this skill ensures that you can accurately and efficiently convey the solutions to inequalities and other mathematical problems.

Conclusion

In conclusion, solving compound inequalities like $6x + 8 \leq 20$ or $5 + 4x \geq 33$ involves a systematic approach that includes solving each inequality individually and then combining their solutions based on the logical connective. For the given inequality connected by "or," we found that the solution is $x \leq 2$ or $x \geq 7$, which is represented in interval notation as $(-\infty, 2] \cup [7, \infty)$.

The process of solving compound inequalities reinforces fundamental algebraic skills, such as isolating variables and understanding how operations affect inequality signs. Additionally, it introduces the concept of logical connectives and how they influence the solution set. The use of "or" in a compound inequality means that the solution includes all values that satisfy either inequality, leading to a union of the individual solution sets.

Graphical representation on a number line provides a visual aid that enhances understanding and helps in accurately expressing the solution in interval notation. The number line clearly shows the ranges of values that satisfy the inequality, making it easier to grasp the concept of the solution set.

Mastering compound inequalities is crucial for further studies in mathematics and its applications in various fields. The ability to solve these types of problems demonstrates a solid understanding of algebraic principles and problem-solving strategies. By following the steps outlined in this comprehensive guide, you can confidently tackle compound inequalities and similar mathematical challenges. The skills acquired through solving compound inequalities are transferable to many other areas of mathematics, making this a valuable topic to master.