Solving Inequalities: The First Step Explained

Hey guys! Let's dive into the world of inequalities, specifically focusing on how to tackle the first step when you're faced with one like $5x + 3 < 18$. Knowing this initial move is super important because it sets the stage for solving the entire problem. The correct answer, as we'll see, is A. Subtract 3. But, why is this the right choice? And what's the big picture of solving inequalities? Let's break it down in a way that's easy to understand.

Understanding Inequalities and Their Basic Structure

Okay, so what exactly is an inequality? Think of it as a mathematical statement that compares two values, showing that they're not equal. Instead of the equals sign (=), we use symbols like:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

In our example, $5x + 3 < 18$, we're saying that the expression on the left side, $5x + 3$, is less than the value on the right side, which is 18. Solving an inequality means figuring out the range of values for the variable (in this case, 'x') that make the statement true. The basic structure of solving an inequality is remarkably similar to solving an equation. You perform operations on both sides to isolate the variable, but there's a crucial difference to keep in mind, which we'll get to later. Remember, the goal is always to get 'x' by itself on one side of the inequality. To do this, we work backward through the order of operations (PEMDAS/BODMAS), undoing the operations that are applied to 'x'. This is where the first step becomes crystal clear.

Now, let's look at our specific problem, $5x + 3 < 18$. The term with 'x' is $5x$, and we have two things happening to it: it's being multiplied by 5, and 3 is being added to it. Following the reverse order of operations, we need to deal with the addition first. That means we need to get rid of the '+ 3'. This is why subtracting 3 is the correct first step.

The Logic Behind Subtracting 3

Why do we subtract 3? Well, it's all about isolating the 'x' term. The equation or inequality is like a balance. Whatever you do to one side, you must do to the other to keep it balanced (or in this case, the inequality true).

So, if we start with $5x + 3 < 18$, we want to get rid of the '+ 3' on the left side. The opposite of adding 3 is subtracting 3. So, we subtract 3 from both sides of the inequality:
$5x + 3 - 3 < 18 - 3$

This simplifies to:
$5x < 15$

See how we've eliminated the constant term (+3) from the left side? Now, our inequality is one step closer to isolating 'x'. This first step, subtracting 3 from both sides, is absolutely essential. It lays the groundwork for the next step, which involves getting rid of the 5 that is multiplying the 'x'.

Let's consider why the other options are not correct. Adding 3 (option B) would make the left side even more complex, not simpler. Multiplying by 5 (option C) would also complicate things, moving us further away from isolating 'x'. Dividing by 5 (option D) is the correct step, but after we've dealt with the addition/subtraction. The order of operations in reverse is key. We first address any addition or subtraction, and then we handle multiplication or division.

The Complete Solution Process

Now that we've taken the first step (subtracting 3 from both sides), let's finish solving the inequality to show the complete picture.
We have: $5x < 15$.
The next step is to isolate 'x'. Currently, 'x' is being multiplied by 5. The opposite of multiplying by 5 is dividing by 5. So, we divide both sides of the inequality by 5:
$\frac{5x}{5} < \frac{15}{5}$
This simplifies to:
$x < 3$

And there you have it! The solution to the inequality $5x + 3 < 18$ is $x < 3$. This means any value of 'x' that is less than 3 will make the original inequality true. For example, if you substitute x = 2 into the original inequality:
$5(2) + 3 < 18$
$10 + 3 < 18$
$13 < 18$

That's true! But, here's a crucial thing to remember. If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you had -2x > 4, you'd divide both sides by -2, and the inequality would become x < -2. This is a common mistake, so keep it in mind! The whole process is all about isolating 'x' and keeping the inequality balanced and true. Now you are one step closer to mastering inequalities.

Practical Examples to Solidify Understanding

To really cement your understanding, let's go through a couple more examples. Practicing different types of inequalities will help you build confidence and recognize the patterns.

Example 1: Solve $2x - 5 \ge 7$

1. First Step: Add 5 to both sides (because we need to undo the subtraction of 5):
   $2x - 5 + 5 \ge 7 + 5$
   This simplifies to:
   $2x \ge 12$
2. Second Step: Divide both sides by 2 (to isolate 'x'):
   $\frac{2x}{2} \ge \frac{12}{2}$
   This simplifies to:
   $x \ge 6$ So, the solution is $x \ge 6$.

Example 2: Solve $-3x + 1 < 10$

1. First Step: Subtract 1 from both sides:
   $-3x + 1 - 1 < 10 - 1$
   This simplifies to:
   $-3x < 9$
2. Second Step: Divide both sides by -3 (and remember to flip the inequality sign because we're dividing by a negative number):
   $\frac{-3x}{-3} > \frac{9}{-3}$
   This simplifies to:
   $x > -3$ So, the solution is $x > -3$.

Notice how in the second example, the inequality sign flipped when we divided by a negative number. This is a crucial detail to remember. By working through these examples, you will see the patterns and become more comfortable with the process.

Common Mistakes and How to Avoid Them

Okay, guys, even the best of us make mistakes! Let's talk about some common pitfalls when solving inequalities, so you can avoid them. One of the biggest errors, as we mentioned earlier, is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Always, always, always be mindful of this rule! It's super important to double-check this step when you are working through your solutions. Another common mistake is not following the order of operations in reverse. Students sometimes jump the gun and try to divide or multiply before dealing with addition or subtraction. Remember to undo the addition or subtraction first! And finally, make sure you perform the same operation on both sides of the inequality. It's easy to get sloppy and accidentally only do it on one side, which will throw off the whole solution.

To avoid these mistakes:

• Write out each step clearly. Don't try to do too much in your head.
• Double-check your work. Go back and review each step to make sure you performed the operations correctly.
• Plug in a test value. Once you think you have the solution, choose a number from your solution set (the range of values for x) and plug it back into the original inequality to see if it makes the statement true. If it does, you're on the right track!

Conclusion: Mastering the First Step

So there you have it! The first step in solving the inequality $5x + 3 < 18$ is to subtract 3 from both sides. This might seem simple, but understanding why this is the correct first move is fundamental to solving more complex inequalities. We've covered the basics of inequalities, the importance of the first step, how to solve inequalities step-by-step, common mistakes to avoid, and provided some helpful examples. Remember, the key is to isolate the variable by using inverse operations, always remembering to perform the same operation on both sides of the inequality and to flip the inequality sign when multiplying or dividing by a negative number. Keep practicing, stay mindful of the rules, and you'll be solving inequalities like a pro in no time! Keep up the great work, and good luck!