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

In the realm of mathematics, solving inequalities is a fundamental skill, particularly when dealing with compound inequalities. Compound inequalities involve two or more inequalities connected by logical connectives such as "OR" or "AND." This article delves into a comprehensive exploration of how to solve the compound inequality $-8x + 14 \geq 60$ OR $-4x + 50 < 58$. We will break down the steps, explain the underlying principles, and provide a clear, step-by-step solution to help you master this essential concept.

Understanding Inequalities

Before we dive into the specifics of the given problem, it’s crucial to understand what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). Unlike equations, which have a single solution or a set of discrete solutions, inequalities often have a range of solutions.

Solving inequalities involves isolating the variable on one side of the inequality symbol, much like solving equations. However, there’s a critical difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. This rule is essential for obtaining the correct solution set.

Breaking Down the Compound Inequality

The given compound inequality is $-8x + 14 \geq 60$ OR $-4x + 50 < 58$. This is a disjunction, meaning that the solution set includes all values of $x$ that satisfy either inequality. To solve this compound inequality, we need to solve each inequality separately and then combine the solution sets.

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

1. Isolate the term with x: Start by subtracting 14 from both sides of the inequality:

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

   This simplifies to:

   $-8x \geq 46$

2. Divide by the coefficient of x: Next, divide both sides by -8. Remember, since we are dividing by a negative number, we must reverse the inequality sign:

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

   This simplifies to:

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

3. Simplify the fraction: Reduce the fraction to its simplest form:

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

So, the solution to the first inequality is $x \leq -\frac{23}{4}$.

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

1. Isolate the term with x: Subtract 50 from both sides of the inequality:

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

   This simplifies to:

   $-4x < 8$

2. Divide by the coefficient of x: Divide both sides by -4. Again, because we are dividing by a negative number, we must reverse the inequality sign:

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

   This simplifies to:

   $x > -2$

Thus, the solution to the second inequality is $x > -2$.

Combining the Solution Sets

Now that we have solved both inequalities, we need to combine their solution sets. The compound inequality is connected by "OR," which means we need to find the union of the two solution sets. In other words, we are looking for all values of $x$ that satisfy either $x \leq -\frac{23}{4}$ or $x > -2$.

The solution set for $x \leq -\frac{23}{4}$ includes all numbers less than or equal to $-5.75$. The solution set for $x > -2$ includes all numbers greater than -2. Since these two intervals do not overlap and are connected by "OR," the combined solution set includes all values in both intervals.

Graphical Representation

To better visualize the solution, we can represent the solution sets on a number line.

• For $x \leq -\frac{23}{4}$, we draw a closed circle at $-5.75$ and shade the line to the left, indicating all values less than or equal to $-5.75$.
• For $x > -2$, we draw an open circle at -2 and shade the line to the right, indicating all values greater than -2.

The entire shaded region represents the solution set for the compound inequality.

The Final Solution

The solution to the compound inequality $-8x + 14 \geq 60$ OR $-4x + 50 < 58$ is the union of the two solution sets, which is:

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

This means that any value of $x$ that is less than or equal to $-\frac{23}{4}$ or greater than -2 will satisfy the original compound inequality.

Conclusion

In this article, we have provided a detailed explanation of how to solve compound inequalities, specifically focusing on the example $-8x + 14 \geq 60$ OR $-4x + 50 < 58$. We have broken down the steps into manageable parts, including solving each inequality separately and then combining the solution sets using the logical connective "OR." Understanding these steps and the underlying principles is crucial for mastering inequalities in mathematics.

The solution to the compound inequality is $x \leq -\frac{23}{4}$ or $x > -2$. This result showcases the importance of carefully applying the rules of inequalities, especially when dividing by a negative number and combining solution sets. By following these guidelines, you can confidently solve a wide range of compound inequalities.

This comprehensive guide aims to equip you with the knowledge and skills necessary to tackle similar problems with ease. Remember to always pay attention to the direction of the inequality sign and the logical connectives between inequalities. With practice, solving compound inequalities will become second nature.