Solving Linear Inequalities: A Step-by-Step Guide
Hey math enthusiasts! Today, we're diving into a classic problem: solving the linear inequality 2(3h + 2) < 46. Don't sweat it, guys! We'll break it down into easy-to-follow steps. This is a fundamental concept in algebra, and understanding it will give you a solid base for more complex mathematical problems. Think of inequalities like a seesaw; we need to keep things balanced while isolating the variable. Let's get started!
Understanding the Basics of Linear Inequalities
Before we jump into the solution, let's chat about what a linear inequality really is. In simple terms, it's a mathematical statement that compares two expressions using inequality symbols. Instead of an equals sign (=), you'll see symbols like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). In our case, we have a 'less than' symbol. This means we're looking for all the values of 'h' that make the left side of the inequality smaller than 46.
Linear inequalities are used to represent a range of solutions, unlike linear equations, which typically have a single solution. The solution to an inequality is often represented as a range of values on a number line. For instance, if our solution is h > 3, it means 'h' can be any number greater than 3. You can picture it as an open circle on the number line at 3, with an arrow pointing to the right, showing that all numbers to the right are included in the solution. If the inequality was h ≥ 3, the circle would be filled in to show that 3 itself is also included in the solution.
The cool thing about linear inequalities is that they show you not just one answer, but a whole bunch of answers that fit the criteria. They're super useful in all kinds of real-world scenarios, like budgeting, figuring out distances, or understanding the limits of something. For instance, if you have a budget, an inequality could tell you the maximum amount you can spend, which means you have a range of values within your spending limit. In this instance, solving linear inequalities helps us to determine the possible values that a variable can take, given the constraints of the inequality. So, understanding the core concepts of linear inequalities sets the stage for solving a variety of mathematical problems and real-world applications.
Step-by-Step Solution to the Inequality
Alright, let's roll up our sleeves and solve the inequality 2(3h + 2) < 46. We'll go step-by-step so it's super clear.
Step 1: Distribute. First things first, we need to get rid of those parentheses. We do this by distributing the 2 across the terms inside the parentheses. This means multiplying both 3h and 2 by 2.
So, 2 * 3h becomes 6h, and 2 * 2 becomes 4. Our inequality now looks like this: 6h + 4 < 46.
Step 2: Isolate the Variable Term. Our goal is to get 'h' all by itself. To do this, we need to get rid of that '+ 4'. We do the opposite operation: subtract 4 from both sides of the inequality. This keeps things balanced, just like our seesaw!
So, 6h + 4 - 4 < 46 - 4. This simplifies to 6h < 42.
Step 3: Solve for h. Almost there! Now, we need to isolate 'h' completely. Currently, it's being multiplied by 6. To get rid of that, we do the opposite: divide both sides of the inequality by 6.
So, 6h / 6 < 42 / 6. This simplifies to h < 7.
And there you have it! The solution to the inequality 2(3h + 2) < 46 is h < 7. This means any number less than 7 will satisfy the original inequality. You can plug in values to check; if you substitute a number less than 7 into the original inequality, you'll find that the inequality holds true. For example, if you substitute 0 for h, you get 2(3*0 + 2) < 46 which simplifies to 4 < 46. This shows that the original inequality is correct.
Visualizing the Solution: The Number Line
Let's visualize the solution h < 7 on a number line. This gives us a great visual representation of all the values that satisfy the inequality.
How to Draw the Number Line:
- Draw the Line: Draw a straight line and put some arrows on both ends to show it goes on forever.
- Mark the Critical Point: Mark the number 7 on the number line. This is the point where the inequality changes.
- Use an Open Circle: Since the inequality is 'less than' (<) and not 'less than or equal to' (≤), we use an open circle at 7. This shows that 7 is not included in the solution.
- Shade the Solution: Shade the part of the number line to the left of 7. This represents all the values less than 7. You can put an arrow on the shaded part to show it goes on forever.
Understanding the Number Line:
The number line helps you see all the possible solutions at a glance. Every point on the shaded side of the number line is a valid solution to your inequality. If you were to pick any number to the left of 7 and substitute it for 'h' in the original inequality, the inequality would be true.
The number line is also useful to solve more complex inequalities and represents how the solution changes when you multiply or divide the inequality by a negative number. When you multiply or divide by a negative number, the direction of the inequality sign flips to the opposite direction, changing which side of the critical point is shaded on the number line. Representing the solution on the number line helps you check your answer and helps you understand how the inequality behaves graphically.
Real-World Applications and Practice Problems
Linear inequalities are used in a variety of real-world scenarios, so understanding how to solve them is a super useful skill. They’re used in all sorts of fields, from business and economics to science and engineering. Here are some examples:
- Budgeting: Let's say you have $100 to spend on clothes, and you want to buy some t-shirts that cost $15 each. If 'x' is the number of t-shirts you can buy, the inequality is
15x ≤ 100. Solving for 'x' will tell you the maximum number of t-shirts you can buy without exceeding your budget. - Distance and Speed: If you're driving at a certain speed, you can use inequalities to determine the time it will take you to travel a certain distance, or the maximum distance you can travel within a specific time frame. The general equation is
distance ≤ speed x time. - Manufacturing: In a factory, inequalities can be used to set production limits. For example, a factory might produce no more than a certain amount of goods per day based on their current resources. That can be translated into an inequality to make sure they're meeting their goals.
Practice Problems:
To really get the hang of solving inequalities, practice is key! Here are a few more problems you can try:
- Solve:
4(2x - 1) > 28 - Solve:
-3(y + 2) ≤ 12 - Solve:
(1/2)z + 5 < 10
Try these problems out and double-check your answers. The more you work with these, the easier they'll become. Remember to always distribute, isolate the variable, and watch out for those negative numbers! Practice makes perfect, and with a little bit of effort, you'll be solving linear inequalities like a pro in no time.
Key Takeaways and Conclusion
Awesome work, guys! We've made it through solving 2(3h + 2) < 46. Let's recap what we've learned.
- Step-by-Step Solution: We went through the process of distributing, isolating the variable term, and solving for 'h'.
- Number Line Visualization: We learned how to represent the solution on a number line, which helps us to visualize the possible values of 'h'.
- Real-World Applications: We explored how inequalities are used in everyday situations, from budgeting to manufacturing.
Solving linear inequalities is a fundamental skill that opens the door to understanding more complex algebra concepts and tackling real-world problems. Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this!
Remember, if you ever feel lost, go back to the basic steps and break down the problem. Math can be tricky, but with a systematic approach and enough practice, anyone can master these concepts. Keep up the great work, and you'll become a pro at solving inequalities in no time!