Solving Inequalities: Find 'g' With Ease

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Solving Inequalities: Find 'g' with Ease

Hey everyone, let's dive into solving inequalities! This is a core concept in mathematics, and it's super important to grasp. Today, we're going to break down the inequality "g + 24 ≥ -47" and figure out how to find the value(s) of 'g' that make this statement true. Don't worry, it's not as scary as it might look at first glance. We'll walk through it step-by-step, making sure you understand every part of the process. Think of it like a puzzle – we're just trying to isolate 'g' on one side of the inequality to find the solution. Ready? Let's get started!

Understanding Inequalities

Okay, before we jump into the problem, let's quickly review what inequalities are all about. Basically, an inequality is a mathematical statement that compares two values, showing that they are not equal. Instead of an equals sign (=), we use symbols like:

  • > (greater than)
  • < (less than)
  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)
  • ≠ (not equal to)

In our case, we have "≥," which means "greater than or equal to." This means that the value of "g + 24" must be bigger than or equal to -47. The cool thing about inequalities is that, unlike equations, they usually have multiple solutions, or a range of values, that satisfy the statement. Instead of just one number being the answer, we get a set of numbers. Our goal is to find this set! In this article, we'll work with the inequality "g + 24 ≥ -47", and this is where we need to find what values of "g" make the statement true. Remember, the symbol "≥" means "greater than or equal to". This means that the value of "g + 24" is equal to or bigger than -47. In order to solve the inequality and find out those values of "g", we need to isolate the variable “g”. We do that with a few simple steps, and we’ll go through them in the next section. But first, let’s quickly recap: the main objective is to find the values of "g" that satisfy the given inequality. The strategy we use is to isolate "g" on one side of the inequality sign. Keep this in mind, and you will do great.

Now, let's solve our example. Imagine you have a number line. On this line, you have -47. Any number to the right of -47 is greater than -47. Any number to the left of -47 is less than -47. So, when we get our answer, we are looking for all the values of "g" that fall on or to the right of a specific point on the number line. That specific point is the solution to the inequality!

Solving the Inequality Step-by-Step

Alright, time to get our hands dirty and actually solve the inequality: g + 24 ≥ -47. The key here is to isolate 'g' on one side of the inequality. To do that, we need to get rid of the "+ 24" that's currently hanging out with 'g'. The beauty of inequalities (and equations, for that matter) is that we can perform the same operation on both sides without changing the overall meaning. So, if we subtract 24 from both sides, we're good to go!

Here’s how it looks:

  1. Start with the original inequality: g + 24 ≥ -47
  2. Subtract 24 from both sides: (g + 24) - 24 ≥ -47 - 24
  3. Simplify: g ≥ -71

See? It's that simple! By subtracting 24 from both sides, we've isolated 'g'. Now, we know that 'g' is greater than or equal to -71. This is our solution! It means any number that is -71 or larger will satisfy the original inequality. For example, -71, -70, -60, 0, 1, 100, and so on, all work. Any number smaller than -71, like -72, -80, or -100, won't work.

Let’s break it down further, just to make sure we're all on the same page. The main action we took was to subtract 24 from both sides of the inequality. This is a fundamental principle: whatever operation you perform on one side of the inequality, you must perform on the other side to keep things balanced. Because we subtracted 24 from the left side, the "+ 24" disappeared, leaving us with just "g". On the right side, subtracting 24 from -47 gives us -71. The "≥" symbol remains the same throughout the entire process because we're not multiplying or dividing by a negative number. This part is super important. Remember, if you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. But we didn't do that here, so we didn't have to worry about it this time.

To make sure you understand the concept, let’s go through a few examples of numbers that fit and don’t fit into the inequality. As a reminder, the inequality states that "g ≥ -71”. The solution indicates that "g" can be any number equal to or greater than -71.

  • Valid values for g: -71, -70, -60, 0, 10, 100, 1000… (These are all equal to or greater than -71)
  • Invalid values for g: -72, -73, -100, -200, -1000… (These are all less than -71)

Visualizing the Solution: The Number Line

Okay, guys, let's visualize our solution on a number line. This will help you understand the concept even better. A number line is a straight line with numbers placed at equal intervals. The number line goes on infinitely in both directions, but for our purposes, we'll focus on the area around -71.

  1. Draw a number line: Draw a straight line and mark some numbers around -71. Include -71, and then some numbers to the left (smaller than -71) and to the right (greater than -71). For example: -74, -73, -72, -71, -70, -69, -68.
  2. Mark the solution: Since our solution is g ≥ -71, we'll put a closed circle (also called a filled-in circle) on -71. The closed circle indicates that -71 is included in the solution. This is because our inequality includes "or equal to". If it was just ">", we'd use an open circle.
  3. Shade the line: Shade the number line from -71 to the right (towards the larger numbers). This shaded area represents all the values of 'g' that satisfy the inequality. All numbers in this shaded area, including -71, are part of the solution.

Basically, the number line shows us all the numbers that make the inequality true. The closed circle on -71 means that -71 is a solution, and the shading to the right shows that all numbers greater than -71 are also solutions. This representation helps make the abstract concept of inequalities much more concrete and easy to understand.

Visualizing solutions to inequalities, especially on a number line, is an excellent way to grasp the range of possible answers. The number line gives a clear picture, where the filled circle at -71 signifies that -71 is indeed a solution, and the shaded part to the right shows all values that make the inequality valid. The number line clarifies the concept, providing a simple yet powerful tool for understanding how inequalities work. It's like a map for all valid solutions!

Checking Your Answer

It's always a good idea to check your answer to make sure you didn’t make any mistakes. Let’s pick a few numbers to plug back into the original inequality and see if they work. Remember, our solution is g ≥ -71.

  1. Check a valid solution: Let's pick g = -70. Substitute -70 into the original inequality: -70 + 24 ≥ -47. Simplifying, we get -46 ≥ -47. This is true! Since -46 is greater than -47, our solution works.
  2. Check another valid solution: Let's pick g = 0. Substitute 0 into the original inequality: 0 + 24 ≥ -47. Simplifying, we get 24 ≥ -47. This is also true! 24 is definitely greater than -47.
  3. Check an invalid solution: Let's pick g = -72. Substitute -72 into the original inequality: -72 + 24 ≥ -47. Simplifying, we get -48 ≥ -47. This is false! -48 is not greater than or equal to -47, so our solution is correct because -72 is not included in our answer.

Checking your answer is an incredibly important step when solving any math problem. It helps catch any errors you might have made along the way and reinforces your understanding of the concepts. By substituting the values back into the original inequality, you ensure that the solution you found is indeed accurate. Think of it as a quality check for your work!

Conclusion

So, there you have it, folks! We've successfully solved the inequality g + 24 ≥ -47, and found that g ≥ -71. We walked through the steps, visualized the solution on a number line, and even checked our answer to be sure. Remember that solving inequalities is very similar to solving equations, but there are a few important differences, especially when dealing with negative numbers. Just take it one step at a time, remember the rules, and you'll be a pro in no time!

This simple example provides a great foundation for tackling more complex inequalities. Keep practicing, and you'll become more confident in solving them. Next time you encounter an inequality, remember the steps we covered today: isolate the variable, and remember to check your work! Good luck, and happy solving!