Algebra Problems: Solutions & Explanations

Hey there, algebra enthusiasts! Let's dive into some problems and break them down step-by-step. We're going to tackle some algebra questions, specifically focusing on card problems. This is all about getting those answers and understanding how we got there. So, grab your pencils, and let's get started. We'll be working through exercises 3 (a, b), 4, 5, and 6. This guide is designed to help you not just find the answers, but to truly understand the concepts behind them. It's like having a friendly tutor right here with you, walking through each problem. We'll be using clear language and plenty of examples to make sure everything clicks. Ready to flex those algebra muscles? Let's go!

Problem 3: Solving Equations (a & b)

Alright, let's kick things off with Problem 3, where we'll be getting our hands dirty with solving equations. This is a fundamental skill in algebra, and it's super important to grasp. We're going to approach each part (a and b) carefully, so you can see the logic behind every step. The main goal here is to isolate the variable, which usually means getting it by itself on one side of the equation. We do this by performing the same operations on both sides to keep everything balanced. Remember the golden rule: what you do to one side, you must do to the other. Let's dig in and solve each part of problem 3, so you can build your confidence. Think of this as a warm-up, prepping you for more complex problems later on. Each step is essential, so make sure to follow along, take notes, and ask questions if something isn't clear.

Part a: Solving the Equation

Let's assume the equation in part (a) looks something like this: 2x + 5 = 15. Our main objective is to figure out the value of x. Remember, we want to isolate x on one side of the equation. Here's how we'd break it down:

1. Isolate the term with x: We need to get rid of the + 5 on the left side. We do this by subtracting 5 from both sides of the equation. This gives us 2x + 5 - 5 = 15 - 5, which simplifies to 2x = 10.
2. Solve for x: Now we have 2x = 10. To find the value of x, we need to divide both sides of the equation by 2. This results in x = 10 / 2, which simplifies to x = 5. So, the solution to this part of the problem is x = 5. We've successfully isolated the variable and found its value. See, that wasn't so bad, right?

Part b: Solving the Equation

Now, let's assume the equation in part (b) is like this: 3(x - 2) = 12. This one has a bit more going on, but don't worry, we'll break it down into easy steps.

1. Distribute: First, we need to get rid of the parentheses. We do this by distributing the 3 across the terms inside the parentheses: 3 * x and 3 * -2. This gives us 3x - 6 = 12.
2. Isolate the term with x: Next, we need to get rid of the - 6 on the left side. We do this by adding 6 to both sides of the equation. This gives us 3x - 6 + 6 = 12 + 6, which simplifies to 3x = 18.
3. Solve for x: Finally, we divide both sides by 3 to find the value of x: x = 18 / 3, which simplifies to x = 6. So, the solution for this part of the problem is x = 6. We've successfully solved for x! See how each step builds on the last? It's all about logical progression.

Problem 4: Working with Inequalities

Alright, let's move on to Problem 4. This time, we're diving into the world of inequalities. Inequalities are similar to equations, but instead of an equals sign (=), we have symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). The rules for solving inequalities are mostly the same as for equations, with one crucial exception: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is super important to remember! Think of it like a seesaw; when you make a change, you need to adjust to keep it balanced. Let's tackle an example to make sure we've got it.

Solving a Sample Inequality

Let's say we have the inequality: 4x - 8 > 12. Here's how we would solve it:

1. Isolate the term with x: First, we add 8 to both sides of the inequality to get rid of the - 8. This gives us 4x - 8 + 8 > 12 + 8, which simplifies to 4x > 20.
2. Solve for x: Next, we divide both sides by 4. Since we're dividing by a positive number, we don't need to flip the inequality sign. This gives us x > 20 / 4, which simplifies to x > 5. So, the solution to this inequality is x > 5. This means any number greater than 5 will satisfy the original inequality. You've got it!

Problem 5: Word Problems and Equations

Alright, let's gear up for Problem 5. This is where we put our algebra skills to the test with word problems! Word problems can seem a bit intimidating at first, but don't sweat it. The key is to break them down into smaller, manageable parts. We'll focus on translating the words into mathematical expressions and equations. This involves identifying the unknowns, assigning variables, and setting up the equation based on the information provided. Once the equation is set up, solving it is just like what we did in Problems 3 and 4. Let's work through a few examples to show how it's done. Remember, practice is key, so let's start with a few examples.

Translating Words into Equations

Let's say we have the following word problem: