Solving And Graphing The Inequality -2y > -6 A Step-by-Step Guide

Jul 31, 2025 by Felix Dubois 66 views

Hey guys! Let's dive into solving and graphing the inequality -2y > -6. This is a fundamental concept in mathematics, and understanding it can help you tackle more complex problems later on. We'll break it down step by step, so don't worry if it seems a bit tricky at first.

Understanding Inequalities

Before we jump into the specific problem, let's quickly recap what inequalities are. Unlike equations that show an exact equality between two expressions (using the '=' sign), inequalities show a range of possible values. We use symbols like '>', '<', '≥', and '≤' to represent these ranges.

> means 'greater than'
< means 'less than'
≥ means 'greater than or equal to'
≤ means 'less than or equal to'

So, in our inequality, -2y > -6, we're looking for all the values of 'y' that, when multiplied by -2, result in a number greater than -6.

Steps to Solve the Inequality

Now, let's solve the inequality -2y > -6. The main goal here is to isolate 'y' on one side of the inequality. We do this using similar algebraic operations as we would with equations, but there's a crucial difference to keep in mind.

Divide both sides by -2:

To get 'y' by itself, we need to divide both sides of the inequality by -2. This is where the important difference comes in. Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

So, when we divide -2y > -6 by -2, we get:
```
(-2y) / -2  _<_  (-6) / -2
```
Notice how the '>' sign flipped to '<'.
Simplify:

Now, let's simplify the expression:
```
y < 3
```
This is our solution! It tells us that any value of 'y' that is less than 3 will satisfy the original inequality.

Graphing the Solution

Okay, we've solved the inequality, but how do we visualize this solution? That's where graphing comes in. We'll use a number line to represent all possible values of 'y', and then we'll highlight the part of the number line that corresponds to our solution.

Draw a number line:

Start by drawing a straight line. Mark zero in the middle, and then add some positive and negative numbers to the left and right. Make sure to include the number 3 on your number line, as that's the critical point in our solution.
Mark the critical point:

Since our solution is 'y < 3', we need to mark 3 on the number line. But here's another important detail: because the inequality is strictly less than (not less than or equal to), we use an open circle at 3. This indicates that 3 itself is not included in the solution. If the inequality were 'y ≤ 3', we would use a closed circle to show that 3 is included.
Shade the solution region:

Our solution includes all values of 'y' that are less than 3. That means we need to shade the part of the number line to the left of 3. This shaded region represents all the possible values of 'y' that make the inequality -2y > -6 true.

Testing the Solution

To make sure we've got it right, let's test a couple of values. First, let's pick a value within our solution region, say y = 0. If we plug this into the original inequality:

-2 * 0 > -6
0 > -6

This is true, so y = 0 is indeed a solution.

Now, let's pick a value outside our solution region, say y = 4. If we plug this in:

-2 * 4 > -6
-8 > -6

This is false, so y = 4 is not a solution. This confirms that our solution and graph are correct!

Common Mistakes to Avoid

Inequalities are pretty straightforward once you get the hang of them, but here are a few common mistakes to watch out for:

Forgetting to flip the inequality sign: This is the biggest one! Always remember to flip the inequality sign when multiplying or dividing by a negative number.
Using the wrong type of circle on the graph: Make sure you use an open circle for strict inequalities ('>' or '<') and a closed circle for inequalities that include equality ('≥' or '≤').
Shading the wrong region: Double-check which direction represents the solution. Values greater than are to the right on the number line, and values less than are to the left.

Examples and Practice

To really solidify your understanding, let's look at a couple more examples. Suppose we have the inequality:

Example 1: 3x + 2 ≤ 11

Subtract 2 from both sides:
```
3x ≤ 9
```
Divide both sides by 3:
```
x ≤ 3
```
So, our solution is x ≤ 3. To graph this, we'd put a closed circle at 3 and shade the region to the left, representing all values less than or equal to 3.

Example 2: -4x - 5 < 7

Add 5 to both sides:
```
-4x < 12
```
Divide both sides by -4 (and flip the sign!):
```
x > -3
```
Our solution here is x > -3. On the number line, we'd place an open circle at -3 and shade to the right, showing all values greater than -3.

Practice Problems

Now it's your turn! Try solving and graphing these inequalities:

2y - 1 > 5
-3z + 4 ≤ -2
4a < 16
-x + 7 ≥ 10

The best way to master inequalities is through practice. Work through these problems, and don't be afraid to check your answers and ask for help if you get stuck.

Real-World Applications

Inequalities aren't just abstract math concepts; they show up in all sorts of real-world situations. Think about:

Budgeting: You might have a budget that says you can spend no more than $100 on groceries. That's an inequality!
Speed limits: A speed limit sign tells you the maximum speed you can drive. Again, an inequality!
Age restrictions: Many activities have age restrictions, like needing to be at least 16 to get a driver's license. You guessed it – another inequality!

Understanding inequalities helps you make decisions and solve problems in these kinds of scenarios.

Conclusion

So, there you have it! We've covered how to solve and graph the inequality -2y > -6, and we've explored the broader concept of inequalities in mathematics. Remember the key steps: isolate the variable, flip the sign when multiplying or dividing by a negative number, and use the number line to visualize your solution. Keep practicing, and you'll become an inequality pro in no time!

Key Takeaways:

Inequalities show a range of possible values.
Flip the inequality sign when multiplying or dividing by a negative number.
Use an open circle for strict inequalities and a closed circle for inequalities that include equality.
Practice makes perfect!