Problem 36 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x \leq 3} \\ {y \leq-1} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the rectangular region that is on or below the line \(y=-1\) and on or to the left of the line \(x=3\).

Step 1: Interpret the Inequalities

The given system of inequalities consists of two inequalities, \(x \leq 3\) and \(y \leq -1\). These inequalities can be depicted on a graph. The inequality \(x \leq 3\) means that the value of variable \(x\) can be anything less than or equal to 3. Similarly, the inequality \(y \leq -1\) represents that the value of variable \(y\) is less than or equal to -1.

Step 2: Graph the Inequalities

To graph the inequalities, consider them as equations, i.e., \(x=3\) and \(y=-1\). Draw the line \(x=3\) which is a vertical line passing through the point (3,0) and then shade the region on the left as it includes values less than 3. Then, Plot the line \(y=-1\) which is a horizontal line passing through the point (0,-1) and shade the region below the line as it includes values less than -1.

Step 3: Find the Intersection

The intersection of the two shaded regions represent the solution set to the system of inequalities. The solution is the rectangular region that is on/below the horizontal line \(y=-1\) and to the left of the vertical line \(x=3\).

Key Concepts

Graphing Inequalities

When tackling the graphing of inequalities, you convert inequalities into a graphical form using simple steps. Each inequality gives a boundary line on the graph. For the exercise given, we looked at the inequalities \( x \leq 3 \) and \( y \leq -1 \).

• Start by grappling them as equations. So, \( x = 3 \) is a vertical line, and \( y = -1 \) is a horizontal line.
• These lines divide the graph into regions. The solution to an inequality lies in one of these regions.
• Since the inequalities use "\( \leq \)", the lines themselves are part of the solution. Thus, darken or highlight the lines to indicate inclusion.

Next, shading helps visualize solutions easily. For \( x \leq 3 \), shade everything to the left side of the line because it includes all \( x \) values less than or equal to 3. Similarly, for \( y \leq -1 \), shade below the line to include values less than or equal to \(-1\). This shading helps illustrate which parts of the graph satisfy the inequalities.

Intersection of Inequalities

The key to solving a system of inequalities is finding where their solutions intersect on the graph. Once we've graphed each inequality, the overlapping area of their shaded regions gives us this intersection. Think of it like finding a common ground where both conditions are met.

• For our exercise, once you have shaded the region to the left of \( x = 3 \) and below \( y = -1 \), the intersecting region is where both shadings overlap.
• This intersection is typically a polygon, or sometimes a less traditional shape, defined by boundaries set by the inequalities.
• In this exercise, it forms a rectangular region in the second quadrant, extending infinitely left but bounded above by \(-1\) and right by 3.

This common region represents potential solution sets that meet all conditions posed by the inequalities.

Solution Set of Inequalities

Understanding the solution set of a system of inequalities involves pinpointing the exact areas on a graph where all inequalities are true simultaneously. This area represents all possible solutions. For the inequalities \( x \leq 3 \) and \( y \leq -1 \), we found the solution in a clearly defined section of the coordinate plane.

• The solution set includes every point within the intersection area, and it reflects all the coordinates \((x, y)\) that satisfy both conditions.
• This set is a visual representation. In our case, it's the region to the left of the line \( x = 3 \) and below the line \( y = -1 \).
• To describe the solution set mathematically, you'd express it as a set of ordered pairs such that both inequalities hold true, \[\{(x, y) \mid x \leq 3, y \leq -1\}\].

Seeing this region on the graph not only aids in understanding but also in verifying which points outside this area do not meet the criteria set by the inequalities. This visual map aids students in grasping complex abstract concepts in a tangible manner.