Solving $5x \leq -15$: Inequality And Graphing Guide

Let's dive into the world of inequalities! In this guide, we'll break down how to solve the inequality $5x \leq -15$ and illustrate the solution set on a number line. Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications. We will go step-by-step, ensuring you grasp every aspect of the process. So, let's put on our math hats and get started!

Understanding Inequalities

Before we jump into solving our specific inequality, let's quickly recap what inequalities are and how they differ from equations. Inequalities are mathematical statements that compare two expressions using symbols like less than (<), greater than (>), less than or equal to ($\leq$), and greater than or equal to ($\geq$). Unlike equations, which have a single solution or a set of discrete solutions, inequalities often have a range of solutions. Understanding these symbols and their implications is the first step in mastering inequalities.

The inequality $5x \leq -15$ tells us that the expression $5x$ is less than or equal to -15. Our goal is to find all values of $x$ that make this statement true. This is where the magic of algebraic manipulation comes in. We'll use techniques similar to those used in solving equations, with a slight twist when it comes to multiplying or dividing by negative numbers. Remember, the core principle is to isolate the variable (in this case, $x$) on one side of the inequality.

When dealing with inequalities, it's essential to pay close attention to the direction of the inequality sign. Multiplying or dividing both sides by a negative number requires flipping the sign to maintain the truth of the statement. This is a crucial rule to remember, and we'll see it in action as we solve our inequality. So, let’s move on to the actual solving process and unravel the solution to $5x \leq -15$ together!

Step-by-Step Solution of $5x \leq -15$

Now, let's get down to business and solve the inequality $5x \leq -15$ step by step. Our primary goal here is to isolate $x$ on one side of the inequality. This will give us a clear picture of all the values that satisfy the given condition. We'll use basic algebraic operations, but we'll also highlight the important rule about flipping the inequality sign when necessary.

1. Isolate x: The inequality we are working with is $5x \leq -15$. To isolate $x$, we need to undo the multiplication by 5. We can do this by dividing both sides of the inequality by 5. Since we are dividing by a positive number, we don't need to worry about flipping the inequality sign. Here's how it looks:

   $5x \leq -15$

   $\frac{5x}{5} \leq \frac{-15}{5}$

2. Simplify:

   Now, let's simplify both sides of the inequality:

   $x \leq -3$

And that’s it! We've successfully isolated $x$. The solution to the inequality is $x \leq -3$. This means that any value of $x$ that is less than or equal to -3 will satisfy the original inequality. Understanding this solution is crucial, but to truly grasp it, we need to visualize it. That’s where graphing the solution on a number line comes into play. So, let’s move on to the next section and see how we can represent this solution graphically.

Graphing the Solution on a Number Line

Visualizing the solution to an inequality on a number line is an incredibly powerful way to understand the range of values that satisfy the inequality. For our solution, $x \leq -3$, we need to represent all numbers that are less than or equal to -3. Here’s how we do it:

1. Draw a number line:

   Start by drawing a straight line. Mark zero in the middle and add some positive and negative numbers to both sides. Make sure to include -3 on your number line, as it's a key value in our solution.

2. Locate -3 on the number line:

   Find -3 on your number line. Since our solution includes -3 (because of the “equal to” part of $\leq$), we will use a closed circle (or a filled-in dot) at -3. A closed circle indicates that -3 is part of the solution set. If the inequality were strictly less than (x < -3), we would use an open circle to show that -3 is not included.

3. Shade the correct side:

   The inequality $x \leq -3$ means we want all values of $x$ that are less than or equal to -3. These values are to the left of -3 on the number line. So, we shade the portion of the number line that extends from -3 to the left (towards negative infinity). This shaded region represents all the numbers that satisfy the inequality.

By graphing the solution on a number line, we can clearly see the range of values that make the inequality true. It's a visual representation that complements the algebraic solution. Now, let’s summarize our findings and wrap up our exploration of this inequality.

Conclusion and Key Takeaways

In this comprehensive guide, we've successfully solved the inequality $5x \leq -15$ and graphically represented its solution on a number line. We started by understanding the basics of inequalities and how they differ from equations. Then, we walked through the step-by-step process of isolating $x$ and arriving at the solution $x \leq -3$. Finally, we learned how to visually represent this solution on a number line, using a closed circle at -3 and shading the region to the left.

Key takeaways from this exercise include:

• Isolating the variable: The primary goal in solving inequalities is to isolate the variable on one side.
• Dividing by a positive number: When dividing (or multiplying) both sides of an inequality by a positive number, the direction of the inequality sign remains the same.
• Graphing the solution: A number line provides a visual representation of the solution set, making it easier to understand the range of values that satisfy the inequality.

Understanding inequalities is crucial for more advanced mathematical concepts, and mastering these foundational skills will set you up for success. We encourage you to practice solving various inequalities and graphing their solutions. The more you practice, the more comfortable you'll become with these concepts.

To further enhance your understanding of inequalities and their applications, consider exploring additional resources such as the materials available on Khan Academy. This trusted website offers a wealth of educational content, including detailed explanations and practice exercises, to help you master mathematical concepts. Keep practicing, and you'll become an inequality-solving pro in no time!