Solving 10 ≤ 4x - 2 < 14 A Step-by-Step Interval Guide

Hey guys! Today, we're diving into the world of inequalities and tackling a specific problem: finding the interval for the inequality $10 \leqslant 4x - 2 < 14$. This might seem a bit daunting at first, but don't worry, we'll break it down step-by-step. Inequalities are a fundamental concept in mathematics, and understanding them is crucial for various fields like computer science, engineering, and economics. So, grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into solving our problem, let's quickly recap what inequalities are. Unlike equations, which state that two expressions are equal, inequalities express a relationship where two expressions are not necessarily equal. We use symbols like $<$ (less than), $>$ (greater than), $\leqslant$ (less than or equal to), and $\geqslant$ (greater than or equal to) to represent these relationships. When we are working with inequalities, it's important to remember that the solution isn't a single value, but rather a range of values. This range is often expressed as an interval, which represents all the numbers that satisfy the inequality. Grasping the basic principles of inequalities is paramount before delving into more complex problems. In essence, an inequality is a mathematical statement that compares two expressions using symbols such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Unlike equations, which seek a specific value that makes the statement true, inequalities aim to find a range or set of values that satisfy the given condition. This distinction is crucial, as it influences the solution process and the representation of the answer. To truly master inequalities, one must understand how operations affect the inequality sign. For instance, adding or subtracting the same value from both sides of an inequality preserves the sign. However, multiplying or dividing by a negative number reverses the inequality sign. This seemingly simple rule is a common pitfall for many, and careful attention must be paid to the sign when performing such operations. Visualizing inequalities on a number line can be immensely helpful. Each inequality represents a region on the number line, and the solution to a compound inequality, like the one we're tackling today, is the intersection or union of these regions. For example, $x > 2$ represents all numbers to the right of 2, while $x \leqslant 5$ represents all numbers to the left of 5, including 5 itself. The ability to translate between algebraic notation and graphical representation is a valuable skill in inequality solving. Moreover, the context of the problem often dictates the type of solution we seek. In some cases, we might be interested in only integer solutions, while in others, any real number within the interval is valid. Understanding the domain of the variable is crucial in interpreting the solution and presenting it appropriately. In practical applications, inequalities arise in various scenarios, such as optimizing resource allocation, determining feasible regions in linear programming, and modeling constraints in real-world problems. The ability to solve inequalities is not just a theoretical exercise but a valuable tool for decision-making and problem-solving in numerous fields.

Breaking Down the Problem: $10 \leqslant 4x - 2 < 14$

Now, let's focus on our specific problem: $10 \leqslant 4x - 2 < 14$. This is a compound inequality, which means it's essentially two inequalities combined into one. We have $10 \leqslant 4x - 2$ and $4x - 2 < 14$. Our goal is to isolate $x$ in the middle, just like we would solve a regular equation. The key here is to perform the same operation on all three parts of the inequality: the left side, the middle, and the right side. To solve the compound inequality $10 \leqslant 4x - 2 < 14$, we need to isolate the variable $x$ in the middle. This involves performing a series of algebraic operations on all three parts of the inequality: the left side, the middle expression, and the right side. The first step is to eliminate the constant term that is being subtracted from the term containing $x$. In this case, we have $-2$, so we add 2 to all three parts of the inequality. This ensures that we maintain the balance of the inequality, just as we would in an equation. Adding 2 to all parts gives us: $10 + 2 \leqslant 4x - 2 + 2 < 14 + 2$, which simplifies to $12 \leqslant 4x < 16$. The next step is to isolate $x$ by dividing all parts of the inequality by the coefficient of $x$. In this case, the coefficient is 4. Dividing all parts by 4 gives us: $\frac{12}{4} \leqslant \frac{4x}{4} < \frac{16}{4}$, which simplifies to $3 \leqslant x < 4$. This is our solution! It tells us that $x$ is greater than or equal to 3, but strictly less than 4. To interpret this solution, we can visualize it on a number line. The solution includes all numbers between 3 and 4, including 3 but not including 4. We represent this interval using bracket notation as $[3, 4)$. The square bracket on the 3 indicates that 3 is included in the solution, while the parenthesis on the 4 indicates that 4 is not included. Understanding the different types of brackets and their meaning is crucial for correctly expressing the solution set of an inequality. Square brackets [ ] indicate that the endpoint is included in the interval, while parentheses ( ) indicate that the endpoint is not included. This distinction is particularly important when dealing with inequalities that involve