Solving The Inequality D-6<-0.5: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of inequalities with a specific problem: d - 6 < -0.5. Inequalities are like equations, but instead of an equals sign, they use symbols like 'less than' (<), 'greater than' (>), 'less than or equal to' (≤), or 'greater than or equal to' (≥). Solving them means finding all the values of the variable (in this case, d) that make the inequality true. So, let's break down this problem step by step, making it super easy to understand, and we'll even graph the solution to visualize it perfectly. Get ready to boost your math skills!
Understanding Inequalities
Before we jump into solving d - 6 < -0.5, let's quickly recap what inequalities are all about. Inequalities compare two values, showing that one is either greater than, less than, greater than or equal to, or less than or equal to the other. The basic inequality symbols you'll encounter are:
- < : Less than
- > : Greater than
- ≤ : Less than or equal to
- ≥ : Greater than or equal to
When we solve an inequality, our goal is to isolate the variable on one side, just like solving an equation. The solution represents a range of values that satisfy the inequality, not just a single value. Also, remember that multiplying or dividing by a negative number flips the inequality sign.
For example, if you have -x < 3, dividing both sides by -1 gives you x > -3. Keep this in mind as we work through our problem!
Step-by-Step Solution of d-6 < -0.5
Okay, let's tackle the inequality d - 6 < -0.5. Our mission is to get d all by itself on one side of the inequality. Here's how we do it:
Step 1: Isolate the Variable
To isolate d, we need to get rid of the -6 on the left side. We can do this by adding 6 to both sides of the inequality. Remember, whatever we do to one side, we must do to the other to keep the inequality balanced.
d - 6 + 6 < -0.5 + 6
This simplifies to:
d < 5.5
Step 2: Interpret the Solution
So, we've found that d < 5.5. This means that any value of d that is less than 5.5 will make the original inequality true. For example, 5, 4, 0, -1, and -100 all satisfy this inequality. However, 5.5, 6, or 10 do not.
Graphing the Solution
Now, let's visualize our solution by graphing it on a number line. Graphing inequalities helps to see the range of values that satisfy the inequality.
Step 1: Draw a Number Line
Start by drawing a straight line. Mark zero in the middle, and then add some numbers to the left and right of zero. Make sure to include the value 5.5 on your number line. Since we're dealing with d < 5.5, we need numbers around 5.5, such as 4, 5, 6, and so on.
Step 2: Place an Open Circle
Because our inequality is d < 5.5 (and not d ≤ 5.5), we use an open circle at 5.5. An open circle indicates that 5.5 itself is not included in the solution. If it were d ≤ 5.5, we would use a closed (filled-in) circle to show that 5.5 is part of the solution.
Step 3: Shade the Line
Since d is less than 5.5, we need to shade the part of the number line to the left of the open circle. This shading represents all the values that are less than 5.5 and, therefore, satisfy the inequality. Add an arrow at the end of your shaded line to show that it continues infinitely in that direction.
The Graph
Your graph should look like this:
- A number line with 0 in the middle.
- An open circle at 5.5.
- The line to the left of 5.5 shaded, with an arrow pointing left.
Examples and Practice Problems
To solidify your understanding, let's go through a few more examples and practice problems.
Example 1: Solve and Graph x + 3 > 7
- Solve: Subtract 3 from both sides: x > 4
- Graph: Draw a number line, place an open circle at 4, and shade to the right.
Example 2: Solve and Graph 2y ≤ 10
- Solve: Divide both sides by 2: y ≤ 5
- Graph: Draw a number line, place a closed circle at 5, and shade to the left.
Practice Problem 1: Solve and Graph m - 2 < 1
Practice Problem 2: Solve and Graph 3n ≥ 9
(Solutions for the practice problems are at the end of this article!)
Common Mistakes to Avoid
When working with inequalities, there are a few common mistakes that students often make. Let's make sure you're not one of them!
- Forgetting to Flip the Inequality Sign: This is crucial! If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you have -2x < 6, dividing by -2 gives you x > -3 (the sign flips!).
- Using the Wrong Type of Circle: Remember, an open circle means the number is not included in the solution ( < or >), while a closed circle means it is included (≤ or ≥).
- Shading in the Wrong Direction: Always double-check which direction to shade. If your solution is x > 3, you shade to the right of 3. If it's x < 3, you shade to the left.
- Not Checking Your Answer: A good practice is to pick a number within your solution range and plug it back into the original inequality to make sure it works. For example, if you solved x > 4, try plugging in 5: is 5 + 3 > 7? Yes, it is! So your solution is likely correct.
Real-World Applications of Inequalities
You might be wondering,