Solving $-15x + 60 \leq 105$ And $14x + 11 \leq -31$ A Step-by-Step Guide

Jul 11, 2025 by ADMIN 74 views

$Solving Compound Inequalities A Comprehensive Guide to $-15x + 60 \_leq 105$ AND $14x + 11 \_leq -31$$

In the realm of mathematics, particularly in algebra, compound inequalities play a crucial role in defining solution sets that satisfy multiple conditions simultaneously. These inequalities involve two or more inequality statements connected by logical operators such as "AND" or "OR." Solving compound inequalities requires a systematic approach, ensuring that each inequality is addressed individually and the combined solution set is accurately determined. This article delves into the process of solving a specific compound inequality, providing a step-by-step guide and offering insights into the underlying concepts.

Understanding Compound Inequalities

Before diving into the solution, it's essential to grasp the fundamental concepts of compound inequalities. A compound inequality linked by "AND" requires that both inequalities must be true for a value to be part of the solution set. Conversely, a compound inequality linked by "OR" requires that at least one of the inequalities must be true for a value to be included in the solution set. The solution set of a compound inequality is the intersection (for "AND") or union (for "OR") of the solution sets of the individual inequalities.

The Given Compound Inequality

Our focus is on the compound inequality: $-15x + 60 \leq 105$ AND $14x + 11 \leq -31$ . This compound inequality presents two linear inequalities connected by the logical operator "AND." To find the solution, we must solve each inequality separately and then determine the intersection of their solution sets.

Step-by-Step Solution

Solving the First Inequality: $-15x + 60 \leq 105$

Isolate the term with x: To begin, we need to isolate the term containing the variable x. This is achieved by subtracting 60 from both sides of the inequality:

$-15x + 60 - 60 \leq 105 - 60$

This simplifies to:

$-15x \leq 45$
Solve for x: Next, we solve for x by dividing both sides of the inequality by -15. It is crucial to remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed:

$\frac{-15x}{-15} \geq \frac{45}{-15}$

This yields:

$x \geq -3$

Thus, the solution to the first inequality is all values of x that are greater than or equal to -3.

Solving the Second Inequality: $14x + 11 \leq -31$

Isolate the term with x: Similar to the first inequality, we isolate the term containing x by subtracting 11 from both sides:

$14x + 11 - 11 \leq -31 - 11$

This simplifies to:

$14x \leq -42$
Solve for x: Now, we solve for x by dividing both sides by 14:

$\frac{14x}{14} \leq \frac{-42}{14}$

This results in:

$x \leq -3$

Therefore, the solution to the second inequality is all values of x that are less than or equal to -3.

Determining the Combined Solution

Now that we have solved each inequality individually, we need to find the intersection of their solution sets. The first inequality, $x \geq -3$ , includes all values of x that are -3 or greater. The second inequality, $x \leq -3$ , includes all values of x that are -3 or less. Since the compound inequality is connected by "AND", we need to find the values of x that satisfy both conditions simultaneously.

The only value that satisfies both $x \geq -3$ and $x \leq -3$ is $x = -3$ . Any value greater than -3 will not satisfy the second inequality, and any value less than -3 will not satisfy the first inequality.

Analyzing the Solution

Graphical Representation

A visual representation can aid in understanding the solution. On a number line, the solution to $x \geq -3$ is represented by a closed circle at -3 and a line extending to the right. The solution to $x \leq -3$ is represented by a closed circle at -3 and a line extending to the left. The intersection of these two solution sets is the single point -3, illustrating that $x = -3$ is the only solution.

Verification

To ensure the correctness of our solution, we can substitute $x = -3$ back into the original compound inequality:

For the first inequality:

$-15(-3) + 60 \leq 105$

$45 + 60 \leq 105$

$105 \leq 105$ (True)

For the second inequality:

$14(-3) + 11 \leq -31$

$-42 + 11 \leq -31$

$-31 \leq -31$ (True)

Since $x = -3$ satisfies both inequalities, it confirms our solution.

Common Mistakes and How to Avoid Them

Forgetting to Reverse the Inequality Sign: A common error occurs when dividing or multiplying an inequality by a negative number. Remember to reverse the direction of the inequality sign in such cases.
Misinterpreting "AND" and "OR": Confusing the logical operators "AND" and "OR" can lead to incorrect solution sets. "AND" requires both inequalities to be true, while "OR" requires at least one to be true.
Incorrectly Combining Solution Sets: Ensure that the solution sets are combined correctly. For "AND", find the intersection, and for "OR", find the union.
Arithmetic Errors: Simple arithmetic mistakes can lead to incorrect solutions. Double-check each step to minimize errors.
Not Verifying the Solution: Always verify the solution by substituting it back into the original inequalities. This helps catch any mistakes made during the solving process.

Conclusion

In summary, the compound inequality $-15x + 60 \leq 105$ AND $14x + 11 \leq -31$ has a single solution: $x = -3$ . Solving compound inequalities involves addressing each inequality individually and then combining their solution sets based on the logical operator connecting them. By following a systematic approach, understanding the underlying concepts, and avoiding common mistakes, you can confidently solve compound inequalities and accurately determine their solution sets. This skill is fundamental in algebra and has applications in various mathematical and real-world contexts.

Understanding compound inequalities is crucial for solving complex mathematical problems. When dealing with inequalities such as $-15x + 60 \leq 105$ AND $14x + 11 \leq -31$ , it's important to break down the problem into smaller, manageable steps. This involves isolating the variable in each inequality and then finding the solution set that satisfies both conditions. Solving inequalities often requires careful attention to detail, especially when dealing with negative coefficients, as this necessitates flipping the inequality sign. The solution set represents the range of values for x that make the inequality true, and in the case of compound inequalities connected by "AND", the solution is the intersection of the solution sets of the individual inequalities. This detailed approach ensures accuracy and a clear understanding of the solution process.

When solving linear inequalities, remember that each step should simplify the expression while maintaining the inequality's validity. For the inequality $-15x + 60 \leq 105$ , the initial steps involve subtracting 60 from both sides, resulting in $-15x \leq 45$ . The next crucial step is dividing both sides by -15. Here, it is vital to remember that dividing by a negative number reverses the direction of the inequality. Thus, $x \geq -3$ is the solution for this inequality. Similarly, for the inequality $14x + 11 \leq -31$ , subtracting 11 from both sides gives $14x \leq -42$ . Dividing both sides by 14 yields $x \leq -3$ . These individual solutions must then be combined based on the "AND" condition, which means finding the overlap between the two solution sets. Understanding these fundamental algebraic manipulations is key to correctly solving compound inequalities and determining the final solution set. The process requires a methodical approach, ensuring each operation is performed accurately to arrive at the correct answer.

To effectively tackle compound inequalities, a strong grasp of algebraic principles is essential. The problem at hand, $-15x + 60 \leq 105$ AND $14x + 11 \leq -31$ , demonstrates the need to solve each inequality separately before combining the solutions. For the first inequality, we start by subtracting 60 from both sides, leading to $-15x \leq 45$ . Dividing by -15 (and flipping the inequality sign) gives us $x \geq -3$ . This step is crucial because overlooking the sign change can lead to an incorrect solution. For the second inequality, subtracting 11 from both sides results in $14x \leq -42$ . Dividing by 14 yields $x \leq -3$ . The compound inequality requires us to find the values of x that satisfy both conditions simultaneously. In this case, the only value that satisfies both $x \geq -3$ and $x \leq -3$ is $x = -3$ . Therefore, the solution set consists of this single value. The ability to accurately perform these algebraic manipulations and logically combine solutions is a cornerstone of mathematical problem-solving.

What is the solution to the compound inequality $-15x + 60 \leq 105$ AND $14x + 11 \leq -31$ ?

Input Keyword Repair

Solve the compound inequality: $-15x + 60 \leq 105$ and $14x + 11 \leq -31$ .