Solving Inequalities A Step-by-Step Guide With Example
Hey guys! Today, we're diving into the world of inequalities and breaking down how to solve them. We'll use a specific example to guide us through each step, making it super clear and easy to follow. So, if you've ever felt a bit puzzled by inequalities, you're in the right place. Let's get started!
Understanding the Inequality
Before we jump into the solution, let's take a quick look at what an inequality actually is. Think of it like an equation, but instead of an equals sign (=), we have signs like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Inequalities help us describe situations where two values aren't necessarily equal but have a specific relationship.
In our case, we're dealing with the inequality 7x + 16 < -5(4 - 5x). This means we're looking for all the values of x that make the left side of the expression less than the right side. Sounds like a mission, right? But don't worry, we'll tackle it step by step.
The Given Inequality: 7x + 16 < -5(4 - 5x)
So, our starting point is the inequality 7x + 16 < -5(4 - 5x). The goal here is to isolate x on one side of the inequality to find out what values of x satisfy this statement. To do this, we'll use a series of algebraic steps, just like solving equations, but with a tiny twist we'll talk about later. Remember, the key is to keep both sides of the inequality balanced while we simplify.
Now, let's break down the solution step-by-step to make sure we understand the reasoning behind each move.
Step 1: Applying the Distributive Property
The first thing we need to do is simplify both sides of the inequality. On the right side, we see -5(4 - 5x), which means we need to distribute the -5 across the terms inside the parentheses. Remember, the distributive property says that a(b + c) = ab + ac. So, we multiply -5 by both 4 and -5x.
- -5 * 4 = -20
- -5 * -5x = 25x
This gives us a new inequality:
7x + 16 < -20 + 25x
Why did we do this? Distributing eliminates the parentheses, making it easier to combine like terms and move towards isolating x. It's like decluttering a room before you start organizing – you need to clear the space first!
Why the Distributive Property Matters
The distributive property is a fundamental concept in algebra. It allows us to simplify expressions by multiplying a term outside parentheses with each term inside. It's a crucial tool for solving equations and inequalities, so it's really important to get comfortable with it. Think of it as unlocking a door – it allows you to access and manipulate the terms inside the parentheses.
By applying the distributive property in this step, we've transformed our inequality into a more manageable form. Now, we can move on to the next step, which involves isolating the variable terms on one side.
Step 2: Isolating the Variable Terms
Now that we've distributed and simplified, let's gather all the x terms on one side of the inequality. We have 7x + 16 < -20 + 25x. To get the x terms together, we can subtract 7x from both sides. This keeps the inequality balanced, just like subtracting the same amount from both sides of a balance scale.
Subtracting 7x from both sides:
7x + 16 - 7x < -20 + 25x - 7x
This simplifies to:
16 < -20 + 18x
What we've done here is move the x term from the left side to the right side. This is a common strategy when solving inequalities or equations – we want to get all the terms with the variable on one side and the constant terms on the other.
The Importance of Maintaining Balance
Remember, when working with inequalities (or equations), whatever you do to one side, you must do to the other. This is crucial for maintaining the truth of the statement. Think of it like a seesaw – if you add or remove weight from one side, you need to do the same on the other to keep it balanced. In our case, subtracting 7x from both sides ensures that the inequality remains valid.
Now that we've isolated the x term on the right side, let's move on to isolating the constant terms on the other side.
Step 3: Isolating the Constant Terms
We're making great progress! Our inequality now looks like this: 16 < -20 + 18x. Now, we want to get all the constant terms (the numbers without x) on one side. We have 16 on the left side, and -20 on the right side. To get rid of the -20 on the right, we can add 20 to both sides. Again, we're keeping the inequality balanced by performing the same operation on both sides.
Adding 20 to both sides:
16 + 20 < -20 + 18x + 20
This simplifies to:
36 < 18x
We've successfully moved the constant terms to the left side, leaving us with just the term with x on the right. We're almost there! Now, we just need to isolate x completely.
Why Is Isolating Constants Important?
Isolating the constant terms is a critical step in solving inequalities because it helps us simplify the expression and get closer to finding the value (or range of values) for x. By moving the constants to one side, we're essentially "cleaning up" the equation, making it easier to see the relationship between x and the remaining numbers.
With the constants isolated, we're now ready for the final step: solving for x.
Step 4: Solving for x
We've reached the final stretch! Our inequality is now 36 < 18x. To completely isolate x, we need to get rid of the 18 that's multiplying it. We can do this by dividing both sides of the inequality by 18.
Dividing both sides by 18:
36 / 18 < 18x / 18
This simplifies to:
2 < x
Or, we can rewrite it as:
x > 2
This is our solution! It means that any value of x greater than 2 will satisfy the original inequality. We've successfully solved for x by isolating it on one side of the inequality.
The Golden Rule of Dividing by a Negative Number
Now, here's a super important thing to remember: If you ever need to divide (or multiply) both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if we had -2x < 4, dividing by -2 would give us x > -2 (notice the flipped sign!).
In our case, we divided by a positive number (18), so we didn't need to flip the sign. But always keep this rule in mind – it's a common mistake to forget it!
Putting It All Together
Let's quickly recap the steps we took to solve the inequality 7x + 16 < -5(4 - 5x):
- Distributive Property: We distributed the -5 on the right side to get rid of the parentheses: 7x + 16 < -20 + 25x.
- Isolating Variable Terms: We subtracted 7x from both sides to get the x terms on one side: 16 < -20 + 18x.
- Isolating Constant Terms: We added 20 to both sides to get the constants on the other side: 36 < 18x.
- Solving for x: We divided both sides by 18 to isolate x: 2 < x or x > 2.
So, the solution to our inequality is x > 2. This means any number greater than 2 will make the original inequality true. We did it!
Checking Our Solution
It's always a good idea to check your solution. Pick a number greater than 2 (let's say 3) and plug it back into the original inequality:
7(3) + 16 < -5(4 - 5(3))
21 + 16 < -5(4 - 15)
37 < -5(-11)
37 < 55
This is true! So, our solution seems correct. If we had gotten a false statement, we'd know we made a mistake somewhere and would need to go back and check our work.
Conclusion
Solving inequalities might seem tricky at first, but by breaking it down into clear steps and understanding the reasoning behind each step, it becomes much more manageable. Remember to use the distributive property, isolate variables and constants, and always keep the golden rule of flipping the sign in mind when dividing or multiplying by a negative number.
And most importantly, practice makes perfect! The more you work with inequalities, the more comfortable you'll become with solving them. So, keep practicing, and you'll be an inequality-solving pro in no time! You've got this!
