Solving 5x - 2 > 28 A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of inequalities, specifically focusing on how to solve the inequality 5x - 2 > 28. Inequalities are a fundamental concept in mathematics, and mastering them is crucial for success in algebra and beyond. We'll break down the steps involved in solving this inequality, explain the reasoning behind each step, and show you how to graph the solution set. So, buckle up and let's get started!

Understanding Inequalities

Before we jump into solving our specific inequality, let's take a moment to understand what inequalities are and how they differ from equations. An equation states that two expressions are equal, using the equals sign (=). An inequality, on the other hand, indicates that two expressions are not necessarily equal. Instead, it shows a relationship where one expression is either greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another expression.

In our case, we have the inequality 5x - 2 > 28. This means we're looking for all the values of 'x' that, when substituted into the expression 5x - 2, result in a value greater than 28. This isn't just one specific value, but rather a range of values, which we'll represent using interval notation and a graph.

Why are inequalities important? Well, they show up everywhere! Think about real-world situations where you have limits or constraints. For example, you might have a budget (a maximum amount you can spend), a minimum grade you need to pass a class, or a speed limit while driving. Inequalities help us model and solve these types of problems.

Key Concepts in Solving Inequalities

Solving inequalities is very similar to solving equations, but there's one crucial difference to keep in mind. Let's go over the key concepts:

• Addition and Subtraction Property: You can add or subtract the same number from both sides of an inequality without changing the solution. This is pretty straightforward – it's the same as with equations.
• Multiplication and Division Property (Positive Numbers): You can multiply or divide both sides of an inequality by the same positive number without changing the solution. Again, just like equations.
• Multiplication and Division Property (Negative Numbers): This is the big one! When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is because multiplying or dividing by a negative number flips the number line, changing the order of the values.
• Interval Notation: This is a way to write down a range of numbers. We'll use parentheses () for values that are not included in the solution (when we have > or <), and brackets [] for values that are included (when we have ≥ or ≤). Infinity (∞) and negative infinity (-∞) always use parentheses.
• Graphing on a Number Line: We can visually represent the solution set of an inequality on a number line. We use open circles for values that are not included (>, <) and closed circles for values that are included (≥, ≤). The line is shaded to show the range of solutions.

Now that we've refreshed our understanding of inequalities, let's get back to our problem!

Step-by-Step Solution of 5x - 2 > 28

Okay, guys, let's tackle the inequality 5x - 2 > 28 step-by-step. We'll follow the same principles we use for solving equations, with that one important rule about flipping the inequality sign in mind.

Step 1: Isolate the term with 'x'

Our goal is to get the 'x' by itself on one side of the inequality. To do this, we need to get rid of the -2 that's hanging out with the 5x. Remember the addition property? We can add 2 to both sides of the inequality without changing the solution:

5x - 2 + 2 > 28 + 2

This simplifies to:

5x > 30

Great! Now the term with 'x' is isolated on the left side.

Step 2: Solve for 'x'

Now we need to get 'x' completely by itself. It's currently being multiplied by 5. To undo this multiplication, we'll use the division property. We can divide both sides of the inequality by 5. Since 5 is a positive number, we don't need to worry about flipping the inequality sign:

5x / 5 > 30 / 5

This simplifies to:

x > 6

Step 3: Interpret the Solution

So, what does x > 6 actually mean? It means that any value of 'x' that is greater than 6 will satisfy the original inequality 5x - 2 > 28. For example, 6.1, 7, 10, 100 – they all work! But 6 itself doesn't work, because the inequality is strictly greater than (>) and not greater than or equal to (≥).

Expressing the Solution in Interval Notation

Interval notation is a handy way to write down the set of all numbers that satisfy an inequality. It uses parentheses and brackets to indicate whether the endpoints are included or not, and it uses infinity symbols to represent unbounded intervals.

In our case, x > 6 means all numbers greater than 6, but not including 6 itself. This goes on forever in the positive direction. So, in interval notation, we write the solution as:

(6, ∞)

The parenthesis next to the 6 means 6 is not included in the solution. The parenthesis next to the ∞ (infinity) always indicates that infinity is not included, because infinity isn't a number – it's a concept.

Graphing the Solution Set

Visualizing the solution set on a number line can be really helpful for understanding what the inequality represents.

Here's how we graph x > 6:

1. Draw a number line. You don't need to include every single number, just enough to show the important point (in this case, 6) and the direction of the solution.
2. Locate 6 on the number line. Since x is strictly greater than 6, we use an open circle at 6. This indicates that 6 is not part of the solution set.
3. Shade the number line to the right of 6. This represents all the numbers greater than 6. Draw an arrow at the end to show that the solution continues to infinity.

The graph visually shows that any point on the shaded line is a solution to the inequality. If you were to pick a number on the shaded line and plug it into the original inequality, it would hold true.

Common Mistakes to Avoid

Solving inequalities is generally straightforward, but there are a few common mistakes students often make. Let's go over them so you can steer clear!

• Forgetting to Flip the Sign: This is the most crucial mistake! Remember, if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. Otherwise, you'll get the wrong solution.
• Incorrect Interval Notation: Pay close attention to whether the endpoint should be included or not. Use parentheses () for > and <, and brackets [] for ≥ and ≤. Also, always use parentheses with ∞ and -∞.
• Misinterpreting the Graph: Make sure you use open circles for > and < and closed circles for ≥ and ≤. The shading should go in the correct direction to represent the solution set.
• Not Checking the Solution: It's always a good idea to pick a number from your solution set and plug it back into the original inequality to make sure it works. This helps you catch any errors you might have made.

Real-World Applications of Inequalities

Inequalities aren't just abstract math concepts; they show up in all sorts of real-world situations. Let's look at a few examples:

• Budgeting: Suppose you have a budget of $100 for groceries. If 'x' represents the amount you spend, the inequality would be x ≤ 100. This means you can spend up to $100, but not more.
• Speed Limits: The speed limit on a highway might be 65 mph. If 's' represents your speed, the inequality would be s ≤ 65. You can drive at 65 mph or slower, but not faster.
• Minimum Grade: To pass a class, you might need a final grade of at least 70%. If 'g' represents your final grade, the inequality would be g ≥ 70. You need a 70% or higher to pass.
• Age Restrictions: Many amusement park rides have height or age restrictions. For example, a ride might require riders to be at least 48 inches tall. If 'h' represents height, the inequality would be h ≥ 48.

These are just a few examples, but they illustrate how inequalities help us model and solve real-world problems that involve limits, constraints, and ranges of values.

Practice Makes Perfect

Alright, guys, we've covered a lot of ground! We've learned how to solve the inequality 5x - 2 > 28, express the solution in interval notation, graph the solution set, and avoid common mistakes. But the key to truly mastering inequalities is practice. Try solving more inequalities on your own, and don't hesitate to ask for help if you get stuck.

Here are a few extra practice problems to get you started:

1. 3x + 5 < 14
2. -2x - 1 ≥ 7
3. 4x - 6 ≤ 2x + 8
4. 5(x + 2) > 3x - 4

Remember to show your work, express your answers in interval notation, and graph the solutions. The more you practice, the more comfortable you'll become with solving inequalities.

Conclusion

Solving inequalities is a fundamental skill in mathematics, and it's essential for success in algebra and beyond. By understanding the basic principles, remembering to flip the sign when multiplying or dividing by a negative number, and practicing regularly, you can master inequalities and apply them to real-world situations.

We hope this guide has been helpful in explaining how to solve the inequality 5x - 2 > 28. Keep practicing, keep learning, and you'll be an inequality pro in no time! Good luck, guys!