Solving & Graphing Inequalities: 2x + 19 < 33

Let's break down how to solve the inequality 2x + 19 < 33 and then graph the solution on a number line. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems later on. We'll go through each step in detail, ensuring you understand not just the how, but also the why behind each operation. So, grab a pen and paper, and let's get started!

Understanding Inequalities

Before diving into the specific problem, it's essential to grasp what inequalities are all about. Unlike equations that state two expressions are equal, inequalities compare expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means finding the range of values that make the inequality true. In our case, we want to find all the values of 'x' that satisfy 2x + 19 < 33. Remember, inequalities are used everywhere, from determining budget constraints to optimizing resources. Understanding them is crucial for real-world problem-solving.

The Golden Rules of Inequality Manipulation

When solving inequalities, you can perform operations on both sides, much like solving equations. However, there's one crucial difference: multiplying or dividing by a negative number reverses the inequality sign. This is a critical rule to remember! For instance, if you have -x < 5, multiplying both sides by -1 gives you x > -5. Failing to flip the sign will lead to an incorrect solution. Other than this, you can add, subtract, multiply, and divide (by positive numbers) just like you would with equations. Keeping these rules in mind will help you navigate through inequality problems with confidence.

Step-by-Step Solution

Now, let's solve the inequality 2x + 19 < 33 step-by-step.

Step 1: Isolate the Term with 'x'

Our first goal is to isolate the term containing 'x', which in this case is '2x'. To do this, we need to get rid of the '+ 19' on the left side of the inequality. We can achieve this by subtracting 19 from both sides:

2x + 19 - 19 < 33 - 19

This simplifies to:

2x < 14

Step 2: Solve for 'x'

Now that we have 2x < 14, we need to isolate 'x' completely. Since 'x' is being multiplied by 2, we can undo this by dividing both sides of the inequality by 2:

2x / 2 < 14 / 2

This gives us:

x < 7

So, the solution to the inequality is x < 7. This means any value of 'x' that is less than 7 will satisfy the original inequality.

Step 3: Understanding the Solution

The solution x < 7 tells us that 'x' can be any number less than 7, but not equal to 7. For example, 6.99, 0, -5, and -100 are all valid solutions. However, 7, 7.01, and 8 are not valid solutions. Understanding this range is crucial for accurately representing the solution on a number line.

Graphing the Solution on a Number Line

Now that we've found the solution, let's graph it on a number line. This visual representation will help solidify your understanding of the solution set.

Setting up the Number Line

Draw a straight line and mark several points along it. These points should include numbers around the solution value (in this case, 7). Make sure to include numbers both greater and less than 7 to clearly show the range of the solution. For example, you might mark points at 4, 5, 6, 7, 8, 9, and 10.

Indicating the Solution

Since our solution is x < 7 (x is less than 7), we need to indicate all the numbers on the number line that are less than 7. Here's how:

• Open Circle at 7: Because 'x' is strictly less than 7 and not equal to 7, we use an open circle at the point 7 on the number line. An open circle signifies that 7 is not included in the solution set.
• Arrow to the Left: Draw an arrow starting from the open circle at 7 and extending to the left (towards negative infinity). This arrow indicates that all numbers less than 7 are part of the solution.

Interpreting the Graph

The graph visually represents all the values of 'x' that satisfy the inequality x < 7. The open circle at 7 shows that 7 is not included, and the arrow extending to the left shows that all numbers smaller than 7 are included. This visual aid is a powerful tool for understanding and communicating the solution to an inequality.

Common Mistakes to Avoid

When working with inequalities, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting to Flip the Inequality Sign: As mentioned earlier, this is a crucial rule. Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
• Incorrectly Graphing the Solution: Pay close attention to whether the inequality includes an