Solving Systems Of Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of solving systems of inequalities. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step, making it super easy to follow. So, grab your pencils, and let's get started!

Understanding Inequalities

Before we jump into solving systems, let's quickly recap what inequalities are. Unlike equations, which have definite solutions, inequalities deal with ranges. Think of it like this: instead of saying x equals a specific number, we say x is greater than, less than, greater than or equal to, or less than or equal to a number. These relationships are represented by the symbols >, <, ≥, and ≤, respectively.

Solving inequalities involves finding all the values of the variable that make the inequality true. The solution is often a range of values rather than a single value, representing all numbers that satisfy the given condition. Understanding how to manipulate inequalities—adding, subtracting, multiplying, and dividing—is crucial for simplifying and solving them. Remember that when multiplying or dividing by a negative number, you need to flip the inequality sign to maintain the truth of the statement. Grasping these fundamental rules sets the stage for tackling more complex systems of inequalities.

When working with inequalities, visualizing the solution on a number line can be incredibly helpful. For instance, if x > 3, you'd represent this on a number line with an open circle at 3 (because 3 is not included) and an arrow extending to the right, indicating all numbers greater than 3. Similarly, if x ≤ -2, you'd use a closed circle at -2 (because -2 is included) and an arrow extending to the left, showing all numbers less than or equal to -2. This visual aid not only clarifies the solution set but also simplifies the process of combining solutions from multiple inequalities in a system. Knowing how to interpret and represent inequalities graphically will undoubtedly boost your confidence and accuracy when dealing with these types of problems.

The properties of inequalities dictate how we can manipulate them while preserving their validity. Adding or subtracting the same number from both sides of an inequality doesn't change the inequality's direction. For example, if a > b, then a + c > b + c and a - c > b - c. However, multiplying or dividing both sides by a positive number also leaves the inequality unchanged. If a > b and c > 0, then ac > bc and a/ c > b/ c. But, and this is crucial, if you multiply or divide by a negative number, you must reverse the inequality sign. So, if a > b and c < 0, then ac < bc and a/ c < b/ c. These rules are essential for isolating variables and finding the solution set for any inequality. Always double-check the sign when multiplying or dividing by a negative number to avoid common errors.

Solving the System

Now, let's tackle the system of inequalities you presented:

a) (2x-1)/4 - (4-x)/2 > 3/4 b) (x-1)/2 < (2-x)/3 + 1/2

We'll solve each inequality separately first.

Solving Inequality a) (2x-1)/4 - (4-x)/2 > 3/4

First, we want to get rid of those fractions! Multiply both sides of the inequality by the least common multiple (LCM) of the denominators, which in this case is 4. This gives us:

4 * [(2x-1)/4 - (4-x)/2] > 4 * (3/4)

This simplifies to:

(2x-1) - 2(4-x) > 3

Now, distribute the -2:

2x - 1 - 8 + 2x > 3

Combine like terms:

4x - 9 > 3

Add 9 to both sides:

4x > 12

Finally, divide by 4:

x > 3

So, the solution to the first inequality is x > 3.

Solving Inequality b) (x-1)/2 < (2-x)/3 + 1/2

Again, let's eliminate the fractions. The LCM of 2 and 3 is 6. Multiply both sides of the inequality by 6:

6 * [(x-1)/2] < 6 * [(2-x)/3 + 1/2]

This simplifies to:

3(x-1) < 2(2-x) + 3

Distribute:

3x - 3 < 4 - 2x + 3

Combine like terms:

3x - 3 < 7 - 2x

Add 2x to both sides:

5x - 3 < 7

Add 3 to both sides:

5x < 10

Divide by 5:

x < 2

Therefore, the solution to the second inequality is x < 2.

Combining the Solutions

Now, we need to find the values of x that satisfy both inequalities simultaneously. We have:

x > 3 x < 2

Looking at these two inequalities, we notice something important: there is no number that can be both greater than 3 and less than 2 at the same time. These two conditions are mutually exclusive. Therefore, there is no solution that satisfies both inequalities simultaneously.

Graphical Representation of the Solution

To better understand why there is no solution, let's visualize each inequality on a number line. For x > 3, we draw an open circle at 3 and shade the region to the right, representing all numbers greater than 3. For x < 2, we draw an open circle at 2 and shade the region to the left, representing all numbers less than 2.

When we look for the overlap between these two shaded regions, we find that there is no common area. This visually confirms that there are no values of x that satisfy both inequalities simultaneously. In other words, the solution sets for the two inequalities do not intersect, leading to an empty solution set for the system.

Using a graphical approach can often provide a clearer understanding of the solution, especially when dealing with multiple inequalities. It allows you to visually identify the regions that satisfy each inequality and determine whether there is any overlap, indicating a common solution. This method is particularly useful when the inequalities are more complex, or when dealing with absolute value inequalities, where the solution might be split into multiple intervals.

Importance of Checking for Overlap

In the context of solving systems of inequalities, the concept of overlap is crucial because it determines whether there is a common solution that satisfies all the inequalities simultaneously. Each inequality in the system defines a range of possible values for the variable, and the solution to the system is the intersection of these ranges. If there is no overlap, it means that no value of the variable can satisfy all the inequalities at the same time, resulting in an empty solution set.

When solving inequalities, it's essential to pay close attention to the direction of the inequality signs and the endpoints of the intervals. An open circle on the number line indicates that the endpoint is not included in the solution, while a closed circle indicates that it is included. Furthermore, when combining the solutions of multiple inequalities, it's helpful to visualize them on a number line to identify the regions of overlap. This graphical representation makes it easier to spot any inconsistencies or contradictions in the system, ensuring that you arrive at the correct solution, or correctly identify when no solution exists.

Conclusion

So, the final answer is that there is no solution to this system of inequalities. It's important to remember that not all systems have solutions, and sometimes, like in this case, the conditions contradict each other. Keep practicing, and you'll become a pro at solving these types of problems in no time! Keep an eye out for more mathematical adventures, and I hope this has helped demystify solving systems of inequalities. Happy solving!