Solving $12x - 20 \leq -17 + 9x$: Interval Notation Guide

Let's dive into the world of inequalities! If you've ever felt a bit puzzled about solving inequalities and expressing the solutions using interval notation, you're in the right place. In this guide, we'll break down the process step by step, using the example inequality 12x - 20 le -17 + 9x to illustrate the concepts. Get ready to conquer inequalities with confidence!

Understanding Inequalities

Before we jump into the solution, it’s crucial to understand what inequalities are and why they're important. Unlike equations that show equality between two expressions, inequalities show a range of possible values. Inequalities use symbols such as less than ($<$), greater than ($>$), less than or equal to ($\leq$), and greater than or equal to ($\geq$).

Inequalities are fundamental in various fields, including mathematics, physics, economics, and computer science. They help describe situations where exact values aren't known, but bounds or ranges are. For example, in economics, you might use inequalities to represent budget constraints, showing how much you can spend without exceeding your income. In physics, inequalities can define the range of possible values for physical quantities like temperature or pressure.

Why is it important to understand inequalities? Mastering inequalities allows you to model real-world scenarios more accurately. Think about situations where you need to stay within certain limits—like speed limits while driving or calorie limits when dieting. Inequalities provide the mathematical tools to handle such constraints effectively. Furthermore, solving inequalities is a foundational skill for more advanced mathematical topics like calculus and optimization problems.

Understanding the symbols is key. The symbol $<$ means 'less than,' while $>$ means 'greater than.' The symbols $\leq$ and $\geq$ include the possibility of equality, meaning 'less than or equal to' and 'greater than or equal to,' respectively. Recognizing these symbols and their meanings is the first step in solving inequalities correctly.

In this guide, we'll tackle the inequality 12x - 20 le -17 + 9x. By working through this example, you’ll learn how to manipulate inequalities algebraically, isolate the variable, and express the solution in interval notation. So, let’s get started and unravel the mystery of inequalities!

Step-by-Step Solution of the Inequality 12x - 20 le -17 + 9x

Now, let’s tackle the given inequality: 12x - 20 le -17 + 9x. We will solve this step by step, ensuring each step is clear and easy to follow. The goal is to isolate the variable $x$ on one side of the inequality.

1. Combine Like Terms

The first step in solving any inequality is to simplify both sides by combining like terms. In this case, we want to get all the $x$ terms on one side and the constant terms on the other side. To do this, we'll start by subtracting $9x$ from both sides of the inequality:

12x - 20 - 9x le -17 + 9x - 9x

This simplifies to:

3x - 20 le -17

Next, we'll add $20$ to both sides to isolate the term with $x$:

3x - 20 + 20 le -17 + 20

Which simplifies to:

3x le 3

2. Isolate the Variable

Now that we have 3x le 3, we need to isolate $x$ completely. To do this, we'll divide both sides of the inequality by $3$:

\frac{3x}{3} le \frac{3}{3}

This gives us:

x le 1

So, the solution to the inequality is $x$ is less than or equal to $1$. This means any value of $x$ that is $1$ or less will satisfy the original inequality.

3. Understanding the Solution

The solution x le 1 tells us that the inequality holds true for all values of $x$ that are less than or equal to $1$. It’s crucial to understand what this means in practical terms. For instance, if $x$ is $0$, the inequality holds true. If $x$ is $-1$, it also holds true. However, if $x$ is $2$, the inequality does not hold true, because $2$ is not less than or equal to $1$.

4. Key Takeaways

• Combine Like Terms: Simplify each side of the inequality by grouping similar terms.
• Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, division) to get the variable alone on one side.
• Divide by a Negative: Remember, if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

By following these steps, you can confidently solve a wide range of inequalities. Next, we’ll learn how to express this solution in interval notation, which is a standardized way of representing solution sets.

Expressing the Solution in Interval Notation

Interval notation is a standardized way to represent a set of numbers using intervals. It’s particularly useful when dealing with inequalities because it clearly shows the range of values that satisfy the inequality. In our case, we found that x le 1. Now, let’s express this solution in interval notation.

Understanding Interval Notation

Before we convert our solution, let’s cover the basics of interval notation.

• Brackets and Parentheses: The two primary symbols used in interval notation are brackets [] and parentheses (). A bracket indicates that the endpoint is included in the interval, while a parenthesis indicates that the endpoint is not included.
• Infinity Symbols: We use the infinity symbol $\infty$ to represent unbounded intervals. Since infinity is not a number, it is always enclosed in a parenthesis.
• Combining Intervals: When the solution set consists of multiple intervals, we use the union symbol $\cup$ to combine them.

Converting x le 1 to Interval Notation

Our solution x le 1 includes all numbers less than or equal to $1$. On a number line, this would be represented by a solid line extending from $1$ to the left, including $1$ itself. In interval notation, this is written as:

$(-\infty, 1]$

Let's break this down:

• $(-\infty$ indicates that the interval extends indefinitely to the left (negative infinity). We use a parenthesis because infinity is not a specific number and cannot be included in the interval.
• $1]$ indicates that the interval includes all numbers up to and including $1$. The bracket signifies that $1$ is part of the solution set.

Thus, the interval notation $(-\infty, 1]$ represents all real numbers less than or equal to $1$.

Examples of Interval Notation

To solidify your understanding, let's look at a few more examples:

1. $x > 3$: This includes all numbers greater than $3$, but not $3$ itself. In interval notation, this is written as $(3, \infty)$.
2. x ge -2: This includes all numbers greater than or equal to $-2$. In interval notation, this is written as $[-2, \infty)$.
3. -1 < x le 4: This includes all numbers between $-1$ and $4$, excluding $-1$ but including $4$. In interval notation, this is written as $(-1, 4]$.
4. $x < 2$ or x ge 5: This includes two separate intervals. In interval notation, this is written as $(-\infty, 2) \cup [5, \infty)$. The $\cup$ symbol means “union,” indicating that we are combining the two intervals.

Common Mistakes to Avoid

• Forgetting the Parenthesis for Infinity: Always use parentheses with infinity symbols.
• Using Brackets Incorrectly: Use brackets only when the endpoint is included in the interval.
• Reversing the Order: The lower bound should always come before the upper bound in interval notation.

By understanding these nuances, you can accurately represent solution sets using interval notation.

Visualizing the Solution on a Number Line

Visualizing solutions on a number line is an excellent way to reinforce your understanding of inequalities and interval notation. A number line provides a clear, graphical representation of the solution set, making it easier to grasp the range of values that satisfy the inequality. Let’s visualize the solution to our inequality, x le 1, on a number line.

Drawing the Number Line

1. Draw a Straight Line: Start by drawing a horizontal line. This line represents all real numbers.
2. Mark the Key Value: In our case, the key value is $1$. Place a mark at $1$ on the number line. This point divides the number line into two regions: numbers less than $1$ and numbers greater than $1$.
3. Use a Closed Circle or Bracket: Since our inequality is x le 1, this means $x$ can be equal to $1$. To indicate that $1$ is included in the solution set, we use a closed circle (or a bracket, if you prefer) at the $1$ mark.
4. Shade the Correct Region: The inequality x le 1 means we are interested in all values less than or equal to $1$. Shade the region to the left of $1$ on the number line. This shaded region represents all the numbers that satisfy the inequality.
5. Extend the Shade: Since the solution includes all numbers less than $1$, the shaded region extends indefinitely to the left. You can draw an arrow at the left end of the shaded region to indicate this.

Interpreting the Visual Representation

The number line representation visually confirms that all numbers to the left of $1$, including $1$ itself, are solutions to the inequality. Numbers to the right of $1$ are not solutions.

Examples of Visualizing Other Inequalities

Let’s look at a few more examples to illustrate how different inequalities are visualized on a number line:

1. $x > 3$:

   • Place a mark at $3$.
   • Use an open circle at $3$ (since $3$ is not included).
   • Shade the region to the right of $3$.

2. x ge -2:

   • Place a mark at $-2$.
   • Use a closed circle at $-2$ (since $-2$ is included).
   • Shade the region to the right of $-2$.

3. -1 < x le 4:

   • Place marks at $-1$ and $4$.
   • Use an open circle at $-1$ (since $-1$ is not included).
   • Use a closed circle at $4$ (since $4$ is included).
   • Shade the region between $-1$ and $4$.

4. $x < 2$ or x ge 5:

   • Place marks at $2$ and $5$.
   • Use an open circle at $2$ (since $2$ is not included).
   • Use a closed circle at $5$ (since $5$ is included).
   • Shade the region to the left of $2$ and the region to the right of $5$.

Benefits of Visualizing on a Number Line

• Clarity: Visualizing solutions on a number line provides a clear picture of the solution set.
• Error Detection: It can help you quickly identify mistakes in your algebraic manipulations.
• Understanding Combined Inequalities: For inequalities with “or” or “and” conditions, a number line makes it easier to see the combined solution.

By visualizing inequalities on a number line, you can enhance your understanding and improve your problem-solving skills.

Conclusion

Congratulations! You've successfully navigated the process of solving the inequality 12x - 20 le -17 + 9x and expressing the solution in interval notation. We've covered the key steps, from simplifying the inequality to visualizing the solution on a number line. By mastering these techniques, you’ll be well-equipped to tackle a wide variety of inequality problems.

Remember, the key to success in mathematics is practice. Work through additional examples, and don't hesitate to revisit the concepts we've covered as needed. Understanding inequalities is a crucial skill that will benefit you in many areas of mathematics and beyond.

We hope this guide has been helpful and has clarified any confusion you may have had about solving inequalities and using interval notation. Keep practicing, and you'll become an inequality-solving pro in no time!

For further learning and practice, consider exploring resources like Khan Academy's Algebra I section on inequalities. It’s a fantastic resource for reinforcing your understanding and tackling more complex problems. Happy solving!