Problem 65 - Thinking Mathematically

Question

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}6 x-y \leq 24 \\ 6 x-y>24\end{array}\right.\)

Step-by-Step Solution

Answer

The system of inequalities has no solution.

Step 1: Analyze the inequalities

Looking at the two inequalities, both are formed by the same linear equation \(6x-y\). The first one is a less than or equal to inequality and the second one is a greater than inequality.

Step 2: Compare the inequalities

The first inequality says that \(6x-y\) is less than or equal to 24, while the second one says that \(6x-y\) is greater than 24. This means a single (x,y) pair satisfying both cannot exist because a number cannot be simultaneously less than (or equal to) 24 and greater than 24.

Step 3: Conclusion

Since the solution for both inequalities can't be the same, it means that the system of equations does not have a common solution.

Key Concepts

Systems of Inequalities

When it comes to understanding systems of inequalities, imagine each inequality as a rule on the number line or coordinate plane that defines where the solutions can be. A 'system' simply means we're looking at two or more of these rules at the same time and trying to find where they all agree. For example, if one inequality tells you that you should be to the left of the number 5 on the number line, and another tells you to be to the right of the number 3, their agreement (or 'system solution') would be the segment between numbers 3 and 5.

However, sometimes inequalities give conflicting rules, such as one saying to be left of a number and another saying to be to the right of that same number, leading to a situation where there cannot be a single solution that satisfies both rules. In these cases, we say the system of inequalities has no solution. Understanding these concepts is crucial to solving inequalities without having to resort to graphing every time.

Linear Inequalities

Let's dive deeper into linear inequalities. These are equations where the highest power of the variable is one. They look very similar to linear equations, but instead of an equal sign, they have inequality signs like \<, \leq, \>, \geq\. Solving a linear inequality is like solving a regular linear equation, but with extra attention to the direction of the inequality.

For instance, if you have an inequality like \(2x - 3 \leq 7\), you solve for \(x\) just as you would in an equation \(2x - 3 = 7\). However, you'll end up with a range of answers, rather than just one. The solution to the inequality will be all the values of \(x\) that make the inequality true. Remember to flip the inequality sign if you ever multiply or divide by a negative number, as it reverses the order.

No Solution Systems

Lastly, let's discuss no solution systems. This type of system occurs when the set of points that satisfy one inequality has no overlap with the set of points that satisfy another inequality. In our original exercise, the system of inequalities \(\left\{\begin{array}{l}6x - y \leq 24\ 6x - y > 24\end{array}\right.\) is an example of a no solution system.

The first inequality suggests all points should fall on or below a certain line, while the second suggests all points should fall above that same line. Since no point can be both above and below the line at the same time, the system has no solution. This concept can be counter-intuitive because, unlike equations, where two distinct lines can intersect at a point, inequality lines define regions, and conflicting definitions of a region result in a lack of common ground, or in this case, a lack of a solution.