Solving Inequalities: A Step-by-Step Guide To Finding 'h'

by ADMIN 58 views

Hey there, math enthusiasts! Today, we're diving into the world of inequalities and, specifically, how to solve for a variable. We're going to take a look at how to solve for h in the inequality: -6 ≥ 4h + 22. Don't worry if it seems intimidating at first; we'll break it down into easy-to-follow steps, making sure everyone understands the process. Get ready to flex those math muscles and master the art of inequality solving! This guide is designed to be your go-to resource, offering clear explanations, helpful examples, and practical tips to boost your understanding. Whether you're a student struggling with algebra or simply someone looking to brush up on your skills, we've got you covered. Let's get started and make solving inequalities a breeze! First, let's understand the foundation of what we're doing. Inequalities are mathematical statements that compare two values, showing whether one is greater than, less than, greater than or equal to, or less than or equal to another. Solving an inequality means finding the range of values that satisfy the statement. Unlike equations, which have a single solution, inequalities often have a range of solutions. For example, if we have something like x > 5, it means that x can be any number greater than 5. Now, let's get into the specific problem we're solving: -6 ≥ 4h + 22. Our goal is to isolate the variable 'h' on one side of the inequality sign. This means we're going to perform operations to get 'h' by itself. Remember, whatever we do to one side of the inequality, we must do to the other side to keep it balanced, just like with equations. It's all about keeping things fair and equal (or in this case, showing the relationship between the two sides accurately)!

Step-by-Step Solution

Now, let’s get to the meat of the matter. We'll break down how to solve for 'h' step-by-step, ensuring you grasp each concept and can apply it confidently. We are given the inequality -6 old{\geq} 4h + 22. Our mission is to find the values of 'h' that satisfy this inequality. Let's begin our journey to solving the inequality! Here's how we'll approach it:

  1. Isolate the term with 'h': Our first move is to get the term containing 'h' by itself on one side of the inequality. In this case, that means getting rid of the +22 that is on the same side as 4h. To do that, we need to subtract 22 from both sides of the inequality. This maintains the balance of the inequality. It's a bit like a seesaw – if you add or remove weight from one side, you have to do the same to the other to keep it level.

    So, we have:
    -6 - 22 ≥ 4h + 22 - 22 Simplifying this gives us:
    -28 ≥ 4h

    Remember, the goal here is to get the 'h' term isolated. By subtracting 22 from both sides, we've removed the constant from the right side, moving us closer to our goal. This step is essential because it allows us to focus on the term with the variable.

  2. Solve for 'h': Now, we've got -28 ≥ 4h. Our next step is to solve for 'h'. Currently, 'h' is being multiplied by 4. To get 'h' all alone, we need to perform the opposite operation: divide both sides of the inequality by 4. Keep in mind that when you divide or multiply both sides of an inequality by a negative number, you need to flip the direction of the inequality sign. Since we're dividing by a positive number (4), we don't need to worry about flipping the sign this time. So, we get:

    -28 / 4 ≥ 4h / 4

    Simplifying this gives us: -7 ≥ h

    This tells us that 'h' is less than or equal to -7.
    This is where things become interesting. Our final result tells us that any value of 'h' that is less than or equal to -7 will satisfy the original inequality. This means that -7, -8, -9, -10, and so on, are all valid solutions. We've successfully isolated 'h' and determined the range of values it can have.

  3. Interpreting the Solution: The solution to the inequality is -7 ≥ h, which can also be written as h ≤ -7. This means that any number less than or equal to -7 satisfies the original inequality. On a number line, you would represent this as a closed circle (because -7 is included) at -7, with an arrow extending to the left, indicating all numbers less than -7. This range of values ensures that when substituted back into the original inequality, the statement remains true. It's like finding the secret code that unlocks the solution to the inequality. By understanding what the solution means, you can better apply the knowledge in various mathematical problems and real-life scenarios. For example, let's test it. If we plug in -7 into the original inequality: -6 ≥ 4(-7) + 22. This simplifies to -6 ≥ -28 + 22, which further simplifies to -6 ≥ -6. This is true! Let's try -8: -6 ≥ 4(-8) + 22, which simplifies to -6 ≥ -32 + 22, and then to -6 ≥ -10. This is also true, as -6 is greater than -10. Any value we pick that is less than or equal to -7 will satisfy the original inequality.

Visualizing the Solution

Visualizing the solution on a number line is super helpful for understanding inequalities. Think of a number line as a straight road, with numbers placed in order, increasing from left to right. Our solution, h ≤ -7, tells us that 'h' can be any number to the left of -7, and including -7 itself. Here’s how you’d show it:

  • Draw the number line: Draw a straight line and mark some numbers, making sure to include -7 and some numbers to the left (like -8, -9, -10) and to the right (like -6, -5, -4). Make sure your number line has a clear scale.

  • Mark -7: Place a closed circle (a filled-in dot) at -7. The closed circle indicates that -7 is included in the solution. If the inequality was h < -7 (without the