Graphing Compound Inequalities Solution To (1/4)s - 10 ≥ -15 Or -(1/9)s - 7 ≥ -4

by Scholario Team 81 views

In the realm of mathematics, compound inequalities play a crucial role in defining ranges of values that satisfy multiple conditions simultaneously. These inequalities involve two or more simple inequalities connected by the words "and" or "or." Understanding how to solve and graph compound inequalities is a fundamental skill in algebra and precalculus. In this comprehensive guide, we will delve into the process of graphing the solution to the compound inequality (1/4)s - 10 ≥ -15 or -(1/9)s - 7 ≥ -4. We will break down the steps involved, provide clear explanations, and illustrate the solution graphically. Mastering compound inequalities not only enhances your problem-solving abilities but also lays a solid foundation for more advanced mathematical concepts.

Before we tackle the specific inequality, let's establish a clear understanding of compound inequalities. A compound inequality is essentially a combination of two or more inequalities. These individual inequalities are linked together by either the word "and" or the word "or," each having a distinct implication on the solution set.

"And" Compound Inequalities

When two inequalities are connected by "and," it signifies that both inequalities must be true simultaneously. The solution set for an "and" compound inequality consists of the values that satisfy both inequalities. This intersection of the solution sets is crucial for finding the valid range of values. Graphically, the solution to an "and" compound inequality is represented by the overlapping region of the individual inequality graphs. For instance, if we have x > 2 and x < 5, the solution includes all numbers greater than 2 and less than 5. The overlapping region on the number line between these two conditions is the solution to the compound inequality. Understanding this intersection is key to solving practical problems where multiple conditions must be met concurrently.

"Or" Compound Inequalities

In contrast, an "or" compound inequality states that at least one of the inequalities must be true. The solution set includes all values that satisfy either inequality or both. This union of solution sets broadens the range of possible values. Graphically, the solution to an "or" compound inequality is represented by the combined regions of the individual inequality graphs. For example, if we consider x < -1 or x > 3, the solution encompasses all numbers less than -1 as well as all numbers greater than 3. There is no overlap required; as long as a number satisfies either condition, it is part of the solution. This distinction is vital in applications where fulfilling any one of several conditions is sufficient to meet a requirement.

Now, let's address the given compound inequality: (1/4)s - 10 ≥ -15 or -(1/9)s - 7 ≥ -4. To solve this, we'll tackle each inequality separately and then combine the solutions based on the "or" condition.

Solving the First Inequality: (1/4)s - 10 ≥ -15

To isolate the variable s in the first inequality, we follow these steps:

  1. Add 10 to both sides of the inequality: (1/4)s - 10 + 10 ≥ -15 + 10 (1/4)s ≥ -5
  2. Multiply both sides by 4 to eliminate the fraction: 4 * (1/4)s ≥ 4 * (-5) s ≥ -20

Thus, the solution to the first inequality is s ≥ -20. This means that any value of s that is greater than or equal to -20 will satisfy this inequality. This forms one part of our overall solution, which we will combine with the solution of the second inequality.

Solving the Second Inequality: -(1/9)s - 7 ≥ -4

Next, we solve the second inequality for s:

  1. Add 7 to both sides: -(1/9)s - 7 + 7 ≥ -4 + 7 -(1/9)s ≥ 3
  2. Multiply both sides by -9. Remember, multiplying by a negative number reverses the inequality sign: -9 * (-(1/9)s) ≤ -9 * 3 s ≤ -27

The solution to the second inequality is s ≤ -27. This indicates that any value of s less than or equal to -27 will satisfy the second inequality. Together with the first inequality's solution, we now have two distinct conditions that define the overall solution to the compound inequality.

Combining the Solutions

Since the compound inequality is connected by "or," we combine the solutions of the two inequalities. The solution set includes all values of s that satisfy either s ≥ -20 or s ≤ -27. This means any number that is greater than or equal to -20 or less than or equal to -27 is a valid solution. This union of solutions is a crucial aspect of “or” compound inequalities, as it broadens the scope of values that fulfill the conditions.

To visualize the solution, we'll graph it on a number line. This graphical representation provides an intuitive understanding of the range of values that satisfy the compound inequality. Here’s how we proceed:

  1. Draw a number line. Mark the critical points, which are -27 and -20, on the number line. These points are the boundaries of our solution intervals. The number line serves as a visual aid to clearly represent the range of values that satisfy the inequality.
  2. For s ≥ -20, draw a closed circle (or a filled-in circle) at -20. A closed circle indicates that -20 is included in the solution. Draw an arrow extending to the right, indicating all values greater than -20. This represents one part of our solution set, encompassing all numbers from -20 onwards to positive infinity.
  3. For s ≤ -27, draw a closed circle at -27, indicating that -27 is included in the solution. Draw an arrow extending to the left, indicating all values less than -27. This represents the other part of our solution set, including all numbers from -27 downwards to negative infinity.
  4. The graph will have two distinct shaded regions: one extending from -20 to the right and the other extending from -27 to the left. These two regions together represent the complete solution set of the compound inequality. This graphical representation vividly shows all possible values of s that satisfy the original compound inequality.

In this comprehensive guide, we've walked through the process of graphing the solution to the compound inequality (1/4)s - 10 ≥ -15 or -(1/9)s - 7 ≥ -4. We began by understanding the fundamental concepts of compound inequalities, distinguishing between "and" and "or" conditions. We then solved each inequality separately, obtaining the solutions s ≥ -20 and s ≤ -27. Finally, we combined these solutions based on the "or" condition and represented the solution set graphically on a number line. By mastering these steps, you can confidently solve and graph a wide range of compound inequalities. Understanding compound inequalities is not just an academic exercise; it's a crucial skill with practical applications in various fields, including engineering, economics, and computer science. The ability to solve and interpret these inequalities enhances your analytical skills and provides a solid foundation for more advanced mathematical studies.

To solidify your understanding, try solving and graphing the following compound inequalities:

  1. 2x + 3 < 7 or -3x + 1 < -8
  2. (1/2)y - 5 ≥ -3 and -4y + 2 ≥ -14
  3. 5z - 10 ≤ 0 or -(1/3)z + 4 > 6

By working through these problems, you'll reinforce your skills and gain confidence in solving and graphing compound inequalities. Remember, practice is key to mastery in mathematics.