Solving $5x - 3 \leq 3x + 7$ A Step-by-Step Guide

In mathematics, solving linear inequalities is a fundamental skill with wide-ranging applications. Linear inequalities are mathematical expressions that compare two values using inequality symbols such as less than ($<$), greater than ($>$), less than or equal to ($\leq$), and greater than or equal to ($\geq$). This comprehensive guide will walk you through the process of solving a specific linear inequality, $5x - 3 \leq 3x + 7$, while providing a broader understanding of the underlying concepts and techniques. Our main focus will be to provide a detailed, step-by-step explanation to ensure clarity and comprehension. Mastering these techniques is crucial for anyone studying algebra or related fields. Understanding how to solve linear inequalities is crucial for various applications in mathematics, science, and engineering. These inequalities often arise when modeling real-world constraints and optimization problems.

Understanding Linear Inequalities

Before diving into the solution, let's establish a solid understanding of what linear inequalities are. A linear inequality is a mathematical statement that compares two linear expressions using one of the inequality symbols mentioned earlier. The goal of solving a linear inequality is to find the range of values for the variable that makes the inequality true. This range is often represented as an interval on the number line. For instance, consider the inequality $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is the variable. Solving this inequality involves isolating $x$ on one side to determine its possible values. Unlike linear equations, which typically have a single solution, linear inequalities often have a range of solutions. This range can include all values greater than, less than, or between certain limits. The solutions are typically expressed in interval notation or graphically represented on a number line. When we work with inequalities, some operations require special attention. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. This is a critical rule to remember to avoid errors. Linear inequalities are foundational in many mathematical contexts, including optimization problems, calculus, and real-world modeling. They allow us to describe constraints and limitations, such as resource availability or physical boundaries. Solving them accurately ensures we can make informed decisions based on mathematical analysis.

Step-by-Step Solution of $5x - 3 \leq 3x + 7$

Now, let's solve the linear inequality $5x - 3 \leq 3x + 7$ step by step. This detailed walkthrough will illustrate the key techniques involved in solving linear inequalities, ensuring you understand each step and its rationale. By breaking down the solution into manageable parts, we can better grasp the process and apply it to other similar problems. Remember, the goal is to isolate the variable $x$ on one side of the inequality. This involves using inverse operations to undo the operations performed on $x$. The steps we'll follow include collecting like terms, simplifying the inequality, and representing the solution in various forms. Pay close attention to the order of operations and the rules for manipulating inequalities. Mastering this process will not only help you solve this specific inequality but also equip you with the skills to tackle more complex problems. Each step is designed to bring us closer to isolating $x$ and determining its range of possible values. Let's begin the step-by-step solution to uncover the values of $x$ that satisfy the given inequality.

1. Combine Like Terms

The first step in solving the inequality $5x - 3 \leq 3x + 7$ is to combine like terms. This involves moving all terms containing $x$ to one side of the inequality and all constant terms to the other side. To do this, we will subtract $3x$ from both sides of the inequality. This ensures that we maintain the balance of the inequality while grouping the $x$ terms together. Subtracting $3x$ from both sides gives us: $5x - 3 - 3x \leq 3x + 7 - 3x$. Simplifying this, we get $2x - 3 \leq 7$. This step is crucial as it brings the variable terms together, making it easier to isolate $x$ in the subsequent steps. Now that we have the $x$ terms on one side, we can proceed to isolate $x$ further by addressing the constant term. Combining like terms is a foundational technique in algebra, not just for solving inequalities but also for simplifying expressions and equations. The goal is to create a more manageable expression that allows us to focus on the variable we want to isolate. In the next step, we will move the constant term to the other side of the inequality to continue isolating $x$. Remember, the key to solving inequalities is to perform the same operations on both sides to maintain the balance and arrive at the correct solution.

2. Isolate the Variable Term

After combining like terms, the inequality is now $2x - 3 \leq 7$. The next step is to isolate the variable term, which in this case is $2x$. To do this, we need to eliminate the constant term, $-3$, from the left side of the inequality. We can achieve this by adding $3$ to both sides of the inequality. Adding $3$ to both sides maintains the balance of the inequality and allows us to isolate the term with $x$. Doing so gives us: $2x - 3 + 3 \leq 7 + 3$. Simplifying this, we get $2x \leq 10$. This step is essential because it brings us closer to isolating $x$ by removing the constant term. Now, we have a simplified inequality where the variable term is the only term on one side. This makes it straightforward to solve for $x$ in the next step. Isolating the variable term is a crucial technique in solving any algebraic equation or inequality. It involves using inverse operations to undo the operations that are attached to the variable. By performing the same operation on both sides, we maintain the balance and arrive at a simpler form that allows us to solve for the variable. In the next step, we will divide both sides by the coefficient of $x$ to finally solve for $x$.

3. Solve for x

Having isolated the variable term, we now have the inequality $2x \leq 10$. The final step in solving for $x$ is to divide both sides of the inequality by the coefficient of $x$, which is $2$. Dividing both sides by $2$ will isolate $x$ and give us the solution to the inequality. When dividing (or multiplying) both sides of an inequality by a positive number, the direction of the inequality sign remains unchanged. This is a critical rule to remember when working with inequalities. Dividing both sides by $2$ gives us: $\frac{2x}{2} \leq \frac{10}{2}$. Simplifying this, we get $x \leq 5$. This is the solution to the inequality, indicating that $x$ can be any value less than or equal to $5$. Solving for $x$ involves using inverse operations to isolate the variable. In this case, division is the inverse operation of multiplication, so we divide both sides by the coefficient of $x$. The solution $x \leq 5$ represents a range of values that satisfy the original inequality. We can represent this solution graphically on a number line or express it in interval notation. In the next section, we will discuss how to represent the solution in different forms to gain a comprehensive understanding of the result. Solving for the variable is the ultimate goal in solving inequalities and equations, and this step brings us to the final answer.

Representing the Solution

After solving the inequality $5x - 3 \leq 3x + 7$, we found that $x \leq 5$. However, the solution can be represented in multiple ways to provide a comprehensive understanding. We can express the solution graphically on a number line, in interval notation, or descriptively in words. Each representation offers a unique perspective on the set of values that satisfy the inequality. Understanding these different forms is crucial for effective communication in mathematics and for applying the solution in various contexts. Representing the solution in multiple ways also reinforces the concept and provides a deeper understanding of the solution set. This section will explore these different representations, providing clear examples and explanations. Whether you prefer a visual representation, a concise interval notation, or a descriptive explanation, this section will equip you with the tools to express the solution of a linear inequality effectively. Let's delve into the various ways we can represent the solution $x \leq 5$.

1. Number Line Representation

One of the most intuitive ways to represent the solution $x \leq 5$ is using a number line. A number line is a visual representation of all real numbers, with numbers increasing from left to right. To represent the solution $x \leq 5$ on a number line, we first locate the value $5$ on the number line. Since the inequality includes