Solving Compound Inequalities $-8x + 14 \geq 60$ OR $-4x + 50 < 58$

Compound inequalities, especially those involving the logical connectors "OR", can initially seem daunting. However, by systematically breaking down each inequality and understanding the implications of the "OR" connector, we can arrive at the solution set. This article provides a step-by-step walkthrough of solving the compound inequality $-8x + 14 \geq 60$ OR $-4x + 50 < 58$, ensuring you grasp the underlying concepts and techniques.

In this comprehensive guide, we will navigate the intricacies of compound inequalities. We will learn how to solve each inequality separately and then combine the solutions using the logical "OR" connective. The key to solving inequalities lies in applying algebraic manipulations while preserving the inequality sign, with careful consideration for negative coefficients. We will explore each step meticulously, providing explanations and justifications to illuminate the process. By the end of this article, you will be equipped with the skills and understanding needed to tackle similar compound inequalities with confidence. Whether you are a student grappling with algebraic concepts or simply seeking to refresh your mathematical knowledge, this guide will serve as a valuable resource.

Our initial problem presents a compound inequality that combines two separate inequalities. The presence of the "OR" connective is crucial, as it dictates how we combine the solutions from each individual inequality. The first inequality, $-8x + 14 \geq 60$, involves a variable term, a constant term, and an inequality sign. Our goal is to isolate the variable x on one side of the inequality. We'll achieve this by performing inverse operations, ensuring we maintain the balance of the inequality. The second inequality, $-4x + 50 < 58$, follows a similar structure and will be solved using analogous techniques. However, a critical aspect to remember when dealing with inequalities is the sign change when multiplying or dividing by a negative number, a detail we will emphasize throughout the solution process. By understanding these fundamental principles and applying them methodically, we can demystify compound inequalities and arrive at accurate solutions.

Let's dissect the compound inequality $-8x + 14 \geq 60$ OR $-4x + 50 < 58$ piece by piece. We'll start by tackling the first inequality:

H3: Solving the First Inequality: $-8x + 14 \geq 60$

1. Isolate the term with x: To begin, we need to isolate the term containing x which is $-8x$. We can do this by subtracting 14 from both sides of the inequality:

   $-8x + 14 - 14 \geq 60 - 14$

   This simplifies to:

   $-8x \geq 46$

2. Solve for x: Now, to solve for x, we need to divide both sides of the inequality by -8. Remember the crucial rule: when dividing or multiplying an inequality by a negative number, you must reverse the direction of the inequality sign.

   $\frac{-8x}{-8} \leq \frac{46}{-8}$

   This simplifies to:

   $x \leq -\frac{46}{8}$

3. Simplify the fraction: We can simplify the fraction $-46/8$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

   $x \leq -\frac{23}{4}$

   So, the solution to the first inequality is x is less than or equal to -23/4. This represents all real numbers that are less than or equal to -5.75. On a number line, this solution would be depicted as a ray extending to the left from -23/4, including the point -23/4 itself. Understanding this visual representation is crucial for grasping the overall solution of the compound inequality, especially when combined with the solution of the second inequality. The interplay between the two solutions, connected by the "OR" operator, will ultimately define the final solution set. Thus, a solid understanding of each individual solution is paramount. We will now proceed to solve the second inequality, keeping in mind the ultimate goal of combining the solutions.

H3: Solving the Second Inequality: $-4x + 50 < 58$

1. Isolate the term with x: Similar to the first inequality, we start by isolating the term containing x, which is $-4x$. We achieve this by subtracting 50 from both sides:

   $-4x + 50 - 50 < 58 - 50$

   This simplifies to:

   $-4x < 8$

2. Solve for x: Now, we divide both sides by -4. Again, we must remember to reverse the inequality sign because we are dividing by a negative number:

   $\frac{-4x}{-4} > \frac{8}{-4}$

   This simplifies to:

   $x > -2$

   Thus, the solution to the second inequality is x is greater than -2. This means all real numbers that are strictly larger than -2 satisfy the inequality. On a number line, this would be represented by a ray extending to the right from -2, but not including -2 itself (usually indicated by an open circle or parenthesis). This solution set is distinct from the first one, and the "OR" connector between the two inequalities means we will be looking for the union of these two solution sets. Before moving on to combining the solutions, it is worth pausing to consider the individual solutions graphically. Visualizing these solutions on a number line provides valuable intuition for understanding the combined solution set. The first solution, x ≤ -23/4, includes all numbers to the left of -23/4, while the second solution, x > -2, includes all numbers to the right of -2. The "OR" connective implies that any number belonging to either of these sets will be part of the final solution.

H3: Combining the Solutions with "OR"

The compound inequality uses the word "OR", which means we are looking for values of x that satisfy either the first inequality or the second inequality (or both).

We found that:

• $x \leq -\frac{23}{4}$
• $x > -2$

To combine these solutions, we consider the union of the two solution sets. The first inequality tells us that x can be any number less than or equal to -23/4. The second inequality tells us that x can be any number greater than -2. Since these conditions are connected by "OR", the solution includes all values that satisfy either condition. There is no overlap between these two solution sets. The first set includes all numbers from negative infinity up to and including -23/4, while the second set includes all numbers strictly greater than -2. Therefore, the combined solution simply consists of the union of these two distinct intervals. We can express this combined solution using inequality notation, which is what we have already done: x ≤ -23/4 OR x > -2. Alternatively, we can express the solution using interval notation, which provides a concise way to represent sets of real numbers. This involves using parentheses and brackets to indicate whether endpoints are included or excluded from the set. The negative infinity symbol (-∞) and positive infinity symbol (∞) are also used to represent unbounded intervals. In this case, the interval notation for x ≤ -23/4 is (-∞, -23/4], and the interval notation for x > -2 is (-2, ∞). The union of these two intervals is then expressed as (-∞, -23/4] ∪ (-2, ∞). Understanding the different ways of representing solution sets is crucial for effective communication in mathematics.

Therefore, the solution to the compound inequality $-8x + 14 \geq 60$ OR $-4x + 50 < 58$ is:

$x \leq -\frac{23}{4}$ or $x>-2$

This corresponds to answer choice A.

In summary, solving compound inequalities involves breaking down the problem into smaller, manageable steps. We first solve each inequality individually, applying the rules of algebraic manipulation while paying close attention to the direction of the inequality sign. Then, we combine the individual solutions based on the logical connector present in the compound inequality. In this case, the "OR" connector signifies that the solution set comprises all values that satisfy either one of the inequalities. Visualizing the solutions on a number line can provide valuable intuition and aid in understanding the combined solution set. By mastering these techniques, you will be well-equipped to tackle a wide range of compound inequalities.