Solving Inequalities -5x - 12 < -32 And 6 - 10x > -84 Step-by-Step Guide

Hey everyone! Today, we're diving into the world of inequalities. Don't worry, it's not as scary as it sounds! We're going to break down how to solve two specific inequalities: $-5x - 12 < -32$ and $6 - 10x > -84$. Think of it like solving a puzzle, where we need to find the range of values for 'x' that make these statements true. So, grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into solving, let's quickly recap what inequalities are all about. Unlike equations, which have a single solution (like x = 5), inequalities deal with ranges of solutions. They use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to). When you're looking at inequalities, you're essentially finding all the possible values of a variable that make the inequality true. This means there isn't just one right answer, but rather a whole set of answers.

So, if you see an inequality like x > 3, this means any number greater than 3 will work. This is a crucial concept to grasp because it sets the stage for how we interpret and express our solutions. We're not just looking for one number; we're looking for a whole range of numbers that fit the condition. Think of it like setting the temperature on a thermostat – you're not just setting it to one degree, but establishing a range within which the temperature will be comfortable. With inequalities, we're doing the same thing with numbers, defining a range where the inequality holds true. This concept of a range of solutions is what sets inequalities apart from equations and is fundamental to understanding how to solve them.

Solving the First Inequality: $-5x - 12 < -32$

Isolating the Variable Term

Okay, let's tackle our first inequality: -5x - 12 < -32. The goal here is to get 'x' all by itself on one side of the inequality. Think of it like peeling away the layers of an onion – we need to undo the operations that are affecting 'x', one step at a time. The first layer to peel back is the '-12'. To do this, we perform the opposite operation, which is adding 12 to both sides of the inequality. This is a crucial step, guys, because whatever you do to one side, you must do to the other to maintain the balance, just like in an equation.

So, when we add 12 to both sides, the inequality transforms like this: -5x - 12 + 12 < -32 + 12. Simplifying this, the -12 and +12 on the left side cancel each other out, leaving us with -5x. On the right side, -32 + 12 equals -20. So, now our inequality looks cleaner and more manageable: -5x < -20. We've successfully isolated the term with 'x', which is a big step forward. This process of adding or subtracting the same number from both sides is a fundamental technique in solving inequalities, and it's all about keeping the balance while we work towards isolating our variable.

Solving for x

We're almost there! Now we have -5x < -20. The final step to isolate 'x' is to get rid of the -5 that's multiplying it. This is where things get a little tricky, so pay close attention. To undo multiplication, we divide. But, and this is a big but, when we divide (or multiply) both sides of an inequality by a negative number, we have to flip the inequality sign. This is a golden rule in the world of inequalities, and it's super important to remember. Think of it like this: multiplying or dividing by a negative number essentially reverses the number line, so we need to flip the sign to keep the inequality true.

So, we divide both sides of -5x < -20 by -5. Because we're dividing by a negative number, the '<' sign flips to '>'. This gives us: (-5x) / -5 > (-20) / -5. Simplifying this, the -5s on the left side cancel out, leaving us with just 'x'. On the right side, -20 divided by -5 equals 4. So, our solution is x > 4. This means any number greater than 4 will satisfy the original inequality -5x - 12 < -32. We've successfully solved for 'x', and remember, we flipped the sign because we divided by a negative number. This step is crucial and can often be a point where mistakes happen, so always double-check when you're multiplying or dividing by a negative in inequalities.

Solving the Second Inequality: $6 - 10x > -84$

Isolating the Variable Term (Again!)

Alright, let's move on to our second inequality: 6 - 10x > -84. Just like before, our mission is to isolate 'x' and find the values that make this statement true. Remember, we're peeling back those layers one by one! The first thing we need to deal with is the '+6' on the left side. Even though there's no explicit plus sign, we understand it's a positive 6. To get rid of it, we'll subtract 6 from both sides of the inequality. This maintains the balance and moves us closer to isolating 'x'.

So, we subtract 6 from both sides: 6 - 10x - 6 > -84 - 6. On the left side, the +6 and -6 cancel each other out, leaving us with -10x. On the right side, -84 - 6 equals -90. Now our inequality looks like this: -10x > -90. We've successfully isolated the term containing 'x'. You see, the process is similar to solving equations, but we're just working with a range of values instead of a single solution. Isolating the variable term is a key step in solving any inequality, and it sets us up for the final step of finding the value (or range of values) of x.

Solving for x (Don't Forget the Flip!)

Okay, we're in the home stretch! We have -10x > -90. To get 'x' completely alone, we need to get rid of the -10 that's multiplying it. Just like in our first inequality, we're going to divide both sides, but remember the golden rule: when we divide (or multiply) by a negative number, we must flip the inequality sign. This is super important, guys, because forgetting this step will lead to the wrong answer. It's like driving on the wrong side of the road – you'll end up in a mess!

So, we divide both sides of -10x > -90 by -10. Because we're dividing by a negative number, the '>' sign flips to '<'. This gives us: (-10x) / -10 < (-90) / -10. Simplifying, the -10s on the left cancel out, leaving us with 'x'. On the right side, -90 divided by -10 equals 9. So, our solution is x < 9. This means any number less than 9 will satisfy the original inequality 6 - 10x > -84. We've successfully solved for 'x', remembering to flip the sign because of the negative division. This little detail makes all the difference in getting the correct solution set for our inequality.

Combining the Solutions

Now we've solved both inequalities individually! We found that for the first inequality, $-5x - 12 < -32$, the solution is x > 4. And for the second inequality, $6 - 10x > -84$, the solution is x < 9. But what if we want to find the values of 'x' that satisfy both inequalities at the same time? This is where combining the solutions comes into play.

To visualize this, think of a number line. We have two conditions: 'x' must be greater than 4, and 'x' must be less than 9. If we were to draw this on a number line, we'd mark an open circle at 4 (because 'x' is strictly greater than 4, not equal to), and shade everything to the right. Then, we'd mark an open circle at 9 (because 'x' is strictly less than 9), and shade everything to the left. The overlap between these two shaded regions represents the values of 'x' that satisfy both inequalities.

The overlap is the region between 4 and 9, not including 4 and 9 themselves. So, we can express the combined solution as 4 < x < 9. This is a concise way of saying that 'x' can be any number between 4 and 9, but it cannot be 4 or 9. This concept of finding the intersection of solutions is crucial in many areas of mathematics, and it's a powerful way to understand how multiple conditions can be satisfied simultaneously. By combining our individual solutions, we've painted a complete picture of the values of 'x' that work for both inequalities.

Expressing the Solution in Interval Notation

There's another cool way to express our solution: interval notation. This is a shorthand way of writing sets of numbers, and it's particularly useful when dealing with inequalities. In interval notation, we use parentheses and brackets to indicate whether the endpoints are included in the solution or not. Parentheses '(' and ')' mean the endpoint is not included (like with our '<' and '>' signs), while brackets '[' and ']' mean the endpoint is included (like with '≤' and '≥' signs).

So, for our combined solution 4 < x < 9, we're saying that 'x' is between 4 and 9, but it can't be 4 or 9. In interval notation, this is written as (4, 9). The parentheses tell us that 4 and 9 are not part of the solution, but every number in between them is. It's like setting boundaries on a playing field – the ball can go anywhere inside the lines, but not on the lines themselves.

Interval notation is a handy tool because it's compact and clear. It quickly conveys the range of values that satisfy the inequality or inequalities. It's widely used in higher-level math courses, so getting comfortable with it now will definitely pay off down the road. Think of it as learning a new language – once you get the hang of it, it's a super efficient way to communicate mathematical ideas.

Conclusion: Mastering Inequalities

And there you have it! We've successfully solved the inequalities -5x - 12 < -32 and 6 - 10x > -84, and we've even combined their solutions and expressed them in interval notation. Remember, the key steps are isolating the variable and, crucially, flipping the inequality sign when you multiply or divide by a negative number. Inequalities might seem tricky at first, but with practice, you'll become a pro at solving them. Keep practicing, guys, and you'll be conquering inequalities in no time!