Problem 74 - Algebra and Trigonometry

Question

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: Understanding the Inequalities

We have two inequalities: \(6x - y \leq 24\) and \(6x - y > 24\). Solving these inequalities will give us the possible values for x and y that make each inequality true.

Step 2: Practically Impossible System

A key observation is that it is impossible for both conditions to occur at the same time. For a number to be simultaneously less than or equal to 24 and greater than 24 is a contradiction. This implies no combination for x and y can satisfy both inequalities.

Step 3: Conclusion From Observations

Since there is no set of values for x and y that can satisfy both inequalities at the same time, it can be concluded that the system of inequalities given has no solution.

Key Concepts

No Solution

In mathematics, sometimes a problem can have no solution. This means there is no set of values that can satisfy the given conditions. For a system of inequalities, having no solution occurs when no numbers exist that simultaneously make all the inequalities true. In our exercise, we have two conflicting inequalities:

• One demands the output be less than or equal to 24
• The other demands it be greater than 24

Mathematically speaking, it's impossible for the same number to fulfill both conditions. Therefore, we say that this system of inequalities has no solution. Recognizing when no solution exists is crucial as it prevents unnecessary computations and helps focus on finding applicable solutions.

Contradictory Inequalities

Sometimes, we encounter inequalities that contradict each other. Contradictory inequalities ask for mutually exclusive conditions, making it impossible to satisfy them both. In our example, we have:

• The inequality \(6x - y \leq 24\)
• The second inequality \(6x - y > 24\)

These inequalities stand in direct contrast. If you tried to graph these on a plane, you'd find that no overlapping area exists that satisfies both at once. This is a telltale sign of a contradiction. Understanding such contradictions helps prevent errors in solving these kinds of problems and helps clarify when a problem isn't solvable within given constraints.

Solving Inequalities

Solving inequalities involves finding all possible values that make an inequality true. The process is similar to solving equations but with special rules, especially around multiplication or division by negative numbers. Always remember:

• When you multiply or divide both sides by a negative number, the inequality sign flips.
• Combine like terms whenever possible.
• Check your solution by plugging values back into the original inequalities.

Inequalities have a range of solutions, often displayed on a number line. In systems, multiple inequalities work together, seeking a common set of solutions, but like in our example, sometimes the constraints are too tight, leading to no solution.

Algebra

Algebra provides the tools needed to handle and solve inequalities systematically. By representing problems with symbols and finding relationships between them, algebra equips us to solve a wide range of mathematical issues. With inequalities, algebra is crucial:

• It allows transformation of expressions into inequalities
• It can introduce assumptions or manipulate terms to find solutions
• It can help visualize solutions, such as graphing to check for contradictions

Through algebraic practices, you can identify not only when a solution exists, but also when no solution is possible, as was highlighted in our example. Understanding these principles helps in efficiently approaching problems with the right strategy.