Solving Inequalities: A Step-by-Step Guide

Nov 10, 2025 by Admin 43 views

Hey guys! Today, we're diving into the world of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Unlike equations that have one specific solution, inequalities often have a range of solutions. Let's break down how to solve some common types of inequalities. Understanding how to manipulate and solve inequalities is super useful not just in math class, but also in real-life situations where you need to compare quantities or determine ranges. So, grab your pencils, and let's get started!

Understanding Inequalities

Before we jump into solving, let's make sure we're all on the same page with what inequalities are and how they work. An inequality compares two values, showing that one is less than, greater than, less than or equal to, or greater than or equal to another. For example, a < b means that a is less than b. Similarly, a ≥ b means that a is greater than or equal to b. Inequalities are used everywhere, from figuring out if you have enough money to buy something to determining if a certain speed is safe for driving conditions. The key thing to remember is that when you perform operations on both sides of an inequality, you need to be careful, especially when multiplying or dividing by a negative number, as this flips the direction of the inequality sign. Knowing these basics will make solving inequalities much easier and more intuitive. So, keep these principles in mind as we tackle some examples.

Solving −3x + 6 ≤ 0

Let's kick things off by solving the inequality −3x + 6 ≤ 0. Our goal here is to isolate x on one side of the inequality. First, we need to get rid of the + 6. We can do this by subtracting 6 from both sides of the inequality. This gives us:

−3x + 6 − 6 ≤ 0 − 6

Which simplifies to:

−3x ≤ −6

Now, we need to get x by itself. Since x is being multiplied by -3, we need to divide both sides by -3. But here's a crucial point: when you divide (or multiply) an inequality by a negative number, you have to flip the direction of the inequality sign. So, we get:

x ≥ −6 / −3

Which simplifies to:

x ≥ 2

So, the solution to the inequality −3x + 6 ≤ 0 is x ≥ 2. This means that any value of x that is greater than or equal to 2 will satisfy the original inequality. To check our work, we can pick a value greater than 2, like 3, and plug it back into the original inequality: −3(3) + 6 ≤ 0, which simplifies to -9 + 6 ≤ 0, or -3 ≤ 0, which is true. Therefore, our solution x ≥ 2 is correct. Remember always to reverse the inequality sign when dividing or multiplying by a negative number.

Solving −3x + 6 ≥ 0

Next up, let's tackle the inequality −3x + 6 ≥ 0. Similar to the previous problem, our aim is to isolate x. First, subtract 6 from both sides:

−3x + 6 − 6 ≥ 0 − 6

Which simplifies to:

−3x ≥ −6

Again, we need to divide both sides by -3 to solve for x. And remember, we have to flip the inequality sign because we're dividing by a negative number:

x ≤ −6 / −3

Which simplifies to:

x ≤ 2

So, the solution to the inequality −3x + 6 ≥ 0 is x ≤ 2. This means that any value of x that is less than or equal to 2 will satisfy the original inequality. Let’s check our answer by plugging in a value less than 2, say 0, into the original inequality: −3(0) + 6 ≥ 0, which simplifies to 6 ≥ 0, which is true. So, our solution x ≤ 2 is correct. Keep in mind the importance of flipping the inequality sign when dividing or multiplying by a negative number to maintain the correctness of the solution.

Solving −3x + 6 ≤ 2

Now, let's solve the inequality −3x + 6 ≤ 2. Just like before, we want to isolate x. First, subtract 6 from both sides:

−3x + 6 − 6 ≤ 2 − 6

Which simplifies to:

−3x ≤ −4

Now, divide both sides by -3. Don't forget to flip the inequality sign:

x ≥ −4 / −3

Which simplifies to:

x ≥ 4/3

So, the solution to the inequality −3x + 6 ≤ 2 is x ≥ 4/3. This means that any value of x that is greater than or equal to 4/3 will satisfy the original inequality. To verify, let’s pick a value greater than 4/3, such as 2, and substitute it into the original inequality: −3(2) + 6 ≤ 2, which simplifies to -6 + 6 ≤ 2, or 0 ≤ 2, which holds true. Therefore, our solution x ≥ 4/3 is accurate. Always double-check your work by plugging in a value from your solution range into the original inequality to confirm its validity.

Solving −3x + 6 ≥ 2

Finally, let's solve the inequality −3x + 6 ≥ 2. Again, our goal is to isolate x. Start by subtracting 6 from both sides:

−3x + 6 − 6 ≥ 2 − 6

Which simplifies to:

−3x ≥ −4

Now, divide both sides by -3, and remember to flip the inequality sign:

x ≤ −4 / −3

Which simplifies to:

x ≤ 4/3

So, the solution to the inequality −3x + 6 ≥ 2 is x ≤ 4/3. This means that any value of x that is less than or equal to 4/3 will satisfy the original inequality. To check, let's pick a value less than 4/3, like 1, and plug it into the original inequality: −3(1) + 6 ≥ 2, which simplifies to -3 + 6 ≥ 2, or 3 ≥ 2, which is true. Thus, our solution x ≤ 4/3 is correct. Consistent practice with these types of problems will solidify your understanding and skills in solving inequalities.

Key Takeaways

Alright, let's wrap up what we've learned about solving these inequalities. Remember, the key steps are:

Isolate the term with x by adding or subtracting constants from both sides.
Divide or multiply by the coefficient of x to solve for x. Important: If you multiply or divide by a negative number, flip the inequality sign!
Check your answer by plugging a value from your solution back into the original inequality.

By following these steps, you can confidently solve a wide variety of inequalities. Inequalities are a fundamental concept in algebra, and mastering them will undoubtedly boost your math skills. Keep practicing, and you'll become a pro in no time! If you found this guide helpful, share it with your friends, and let's conquer math together! Keep practicing and you will master the concepts in no time! Good luck!