Problem 82 - Precalculus

Question

In Exercises \(79-84,\) determine whether each statement is true or false. A system of linear inequalities always has a solution.

Step-by-Step Solution

Answer

False; a system of linear inequalities does not always have a solution.

Step 1: Understanding the Statement

Before determining the truth of the statement, let's understand what it implies. A system of linear inequalities involves finding all possible solutions which satisfy all inequalities simultaneously. The statement claims that there is always at least one such solution.

Step 2: Visualizing Systems of Inequalities

Visualize linear inequalities on a graph as regions. For one inequality, the solution is a half-plane. A system of inequalities involves the intersection of these half-planes.

Step 3: Considering Possible Intersections

Two linear inequalities may have an intersecting region, but this isn't guaranteed. For example, if two inequalities represent parallel lines with different shading directions, there can be no intersection, thus no solution.

Step 4: Real-world Example

Consider the system: \( y > 2 \) and \( y < 1 \). There is no value of \( y \) that simultaneously satisfies both inequalities, illustrating a case with no solutions.

Step 5: Conclusion

Since it is possible to create systems of linear inequalities without a solution—such as conflicting inequalities—the statement is proven false.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of equal signs, they use inequality symbols such as "<", ">", "≤", or "≥". These inequalities represent ranges of possible solutions rather than specific values. For example, take the inequality \( y > 2 \). This means any value of \( y \) greater than 2 is considered a solution.

• The direction of the inequality symbol tells us which side of the boundary line on a graph is included in the solution set.
• The boundary itself is represented by a dashed or solid line, depending on whether the inequality symbol allows equality (solid for ≤, ≥ and dashed for <, >).

Linear inequalities can form systems, where multiple inequalities are considered together. Each one adds constraints, narrowing down the region where potential solutions lie.

Graphical Solutions

When dealing with systems of linear inequalities, graphical solutions provide an intuitive way to visualize where solutions exist. Each inequality in the system is graphed in a two-dimensional coordinate system as a half-plane.

• The half-plane is the region on one side of the inequality's boundary line.
• To find solutions, look for areas where all half-planes overlap.

By shading the relevant half-planes on a graph, you can see the common region that satisfies all inequalities at once. If the system of inequalities is consistent, you'll see a distinct overlap. If not, no overlapping region means no solution.

Solution Intersections

The key to finding solutions to a system of linear inequalities is looking for solution intersections. Each half-plane represents potential solutions for a particular inequality.

• Intersections are areas where these planes overlap.
• These intersections form a feasible region that represents the solution to the entire system.

However, intersections aren't always guaranteed. If the inequalities are contradictory, such as one stating \( y > 2 \) and another \( y < 1 \), the lines do not overlap and no solution exists.It's these intersections which determine whether a system has a solution, demonstrating why systems can sometimes have none.

Real-world Examples

Real-world scenarios often require the use of systems of inequalities to model conditions and constraints.For instance:

• A company may want to ensure profit falls above a certain level, while expenses stay below another, resulting in inequalities like \( profit > 1000 \) and \( expenses < 800 \).
• In such situations, the feasible region defined by the system's solution gives insight into reachable goals and helpful constraints.

These practical examples show how systems of inequalities express constraints that can influence decision-making and resource management. They translate abstract math into actionable understanding, aligning complex requirements with available resources.